| Model | Memory Architecture | Attentional Bias | Retention Gate† | Memory Algorithm | Memory Write Operation |
| Shallow Memory |
| RetNet (2023) | Vector | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| Transformer (2017) | Matrix | L2 | - | Nonparametric | Mt=Mt-1∪{kt, vt} |
| LA (2021) | Matrix | Dot-Product | - | GD | Mt=Mt-1+vtktT |
| DFW | Matrix | Dot-Product | L2 | GD | Mt=(βtαT) ⊙ Mt-1+vtktT |
| Lightening Attention (2025) | Matrix | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| GLA (2024) | Matrix | Dot-Product | L2 | GD | Mt=Diag(αt)Mt-1+vtktT |
| Mamba (2024) | Matrix | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| HGRN2 (2024) | Matrix | L1 | L2 | GD | Mt=Diag(αt)Mt-1+vt(1-αt)T |
| DeltaNet (2017) | Matrix | L2 | - | GD | Mt=(I-βtktkT)Mt-1+βtvtktT |
| Longhorn (2024) | Matrix | L2 | - | Implicit GD | Mt=(I-βtktkT)Mt-1+(βt1+ktkβt)xtkT |
| TTT-Linear (2024) | Matrix | L2 | - | GD | Mt=Mt-1-η∇L(Mt-1, xt) |
| Gated DeltaNet (2024) | Matrix | L2 | L2 | GD | Mt=(αt(I-βtktkT))Mt-1+βtvtktT |
| RWKV-7 (2025) | Matrix | L2 | L2 | GD | Mt=diag(αt)(I-βtktkT)Mt-1+βtvtktT |
| DeltaProduct (2025) | Matrix | L2 | L2 | MGD* | Mt=(αtΠi=1n(I-βt,ikt,i)T)Mt-1+Σj=1nΠi=j(I-βt,ivtj,kj,i) |
| Deep Memory |
| TTT-MLP (2024) | 2-layer MLP | L2 | - | GD | Mt=Mt-1-η∇L(Mt-1;kt, vt) |
| Titans-LMM (2024) | k-layer MLP | L2 | L2 | GD + Momentum | Mt=αMt-1-St, where St=ηSt-1-θt∇L(Mt-1;kt, vt) |
| MONETA (ours) | 2-layer MLP | Lp | Lq | GD | At=AtA1-ηt∇lp(Wt-1;kt, vt), Wt=At/||At||q-2 |
| YAAD (ours) | 2-layer MLP | Huber | L2 | GD | Wt=atWt-1-(ηt∇ε2(Wt-1;kt, vt) if ||M(kt)-vt|≤δt, ηtδt∇ε1(Wt-1;kt, vt) Otherwise. |
| MEMORA (ours) | 2-layer MLP | L2 | KL | GD | Wt=Softmax(αt log(Wt-1)-ηt∇ε2(Wt-1;kt, vt)) |
",
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+ "type": "text",
+ "text": "Deep Memory Module: Titans and Test Time Training. To overcome the limited memory and to extend the effective context length of deep sequence models, more recent studies focus on a new generation of architectures with deep memory module (Behrouz et al. 2024c; Sun et al. 2024). These architectures are built on the meta-learning perspective, where the memory is an MLP architecture that is updated using gradient descent (with momentum) (Behrouz et al. 2024c; Sun et al. 2024). Sun et al. (2024) further provide a unifying perspective that how linear and softmax attention are respectively parametric and non-parametric solutions of (kernel) regression loss but consider other modern linear RNNs outside of this class of models. Recently, in a concurrent work to ours, Wang et al. (2025) show that with additional simplification of modern RNNs (e.g., RetNet (Sun et al. 2023), Mamba (Dao et al. 2024)) they approximately place in the same class of models that internally optimize regression loss. It, however, still remains unanswered that \"What is the underlying design framework behind these sequence models that can accurately unify existing architectures?\" Moreover, the role of forget gates and its alternative choices in modern sequence models is surprisingly less explored.",
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+ "type": "text",
+ "text": "3 Associative Memory, Attentional Bias, and Retention",
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+ "text": "Associative memory, which is an inseparable component of learning in humans (Terry 2017), has been the inspiration for many artificial neural architectures in the literature (Behrouz et al. 2024c; Hopfield 1982; Neil et al. 2017). These studies, however, define instances of the concept of associative memory, limiting the architecture to a specific class of similarity metrics between entities (i.e., keys and values). That is, broadly speaking, associative memory is an operator that maps a set of keys $K$ to a set of values $V$ , and so to learn the underlying mapping patterns in data, it requires an objective that targets a type of memory and measures the quality of learned mappings:",
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+ "text": "Definition 3.1 (Associative Memory and Attentional Bias). Given a set of keys $\\mathcal{K} \\subseteq \\mathbb{R}^{d_k}$ and values $\\mathcal{V} \\subseteq \\mathbb{R}^{d_o}$ , associative memory is an operator $\\mathcal{M}: \\mathcal{K} \\to \\mathcal{V}$ . Learning the mapping of associative memory is based on an objective $\\mathcal{L}$ , called",
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+ "type": "page_number",
+ "text": "4",
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+ "type": "text",
+ "text": "Attentional Bias, that determines the type of memory and its tendency to prioritize some events:",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {M} ^ {*} = \\arg \\min _ {\\mathcal {M}} \\quad \\mathcal {L} (\\mathcal {M} (\\mathcal {K}); \\mathcal {V}). \\tag {4}\n$$\n",
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+ "type": "text",
+ "text": "A few remarks are in order:",
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+ "type": "text",
+ "text": "Remark 1. When we parameterize the memory with parameter $W$ , we use $\\mathcal{M}(W, \\mathbf{k})$ . In this parametric setting, the optimization problem in (4) should be performed over the parameter $W$ . Furthermore, in the parametric setup, we might use an additional regularization $\\mathcal{R}(W)$ to control the retaining of the past data.",
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+ "type": "text",
+ "text": "Remark 2. Learning the mapping between keys and values (Equation 4) is a meta-learning problem, in which the attentional bias is optimized in the inner-loop and all other parameters of the neural network (e.g., linear projections, convolutions, etc.) are optimized in the outer-loop. Therefore, the model learns how to store the data into its parameters at test time (Behrouz et al. 2024c; Sun et al. 2024).",
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+ "type": "text",
+ "text": "3.1 Learning to Memorize and to Retain Through the Lens of Optimization",
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+ "text": "Definition 3.1 translates the design of a neural architecture based on the concept of associative memory to learning the underlying mapping between keys and values, by minimizing an objective $\\mathcal{L}$ . To optimize Equation 4, one simple approach is to utilize the idea of gradient descent. Specifically, given a new pair of keys and values, we update the memory as:",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = W _ {t - 1} - \\eta_ {t} \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right), \\tag {5}\n$$\n",
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+ "text": "where, for simplicity, we use the definition $\\ell(W_{t-1}; \\mathbf{k}_t, \\mathbf{v}_t) \\coloneqq \\mathcal{L}(\\mathcal{M}(W; \\mathbf{k}_t), \\mathbf{v}_t)$ . Behrouz et al. (2024c) re-interpret the formulation as a momentary surprise metric, where the model memorizes tokens that violates the expectation of the objective (i.e., being surprising to the memory). Although the choice of objective is an important step to fully interpret Equation 5 (which we discuss in detail in Section 5), there are different viewpoints to interpret this update rule in its general format, which later can help us to go beyond existing architectures:",
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+ "type": "text",
+ "text": "3.2 Viewpoint 1: Online Regression and Follow-The-Regularized-Leader",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Equation (5) can be viewed as one step of online gradient descent over the sequence of the loss functions",
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+ "type": "equation",
+ "text": "\n$$\n\\ell \\left(W; \\mathbf {k} _ {1}, \\mathbf {v} _ {1}\\right), \\ell \\left(W; \\mathbf {k} _ {2}, \\mathbf {v} _ {2}\\right), \\dots , \\ell \\left(W; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right), \\dots . \\tag {6}\n$$\n",
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+ "text": "It is well known that the online gradient descent can be viewed as a special case of Follow-The-Regularized-Leader (FTRL) algorithm with a special choice of loss functions (Shalev-Shwartz et al. 2012, Chapter 2) and (Hazan et al. 2016). Specifically, assuming $W_0 = 0$ , the update rule in (5) is equivalent to",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = \\arg \\min _ {W} \\quad \\sum_ {i = 1} ^ {t} \\left\\langle W - W _ {i - 1}, \\nabla \\ell \\left(W _ {i - 1}; \\mathbf {k} _ {i}, \\mathbf {v} _ {i}\\right) \\right\\rangle + \\frac {1}{2 \\eta} \\| W \\| _ {2} ^ {2}, \\tag {7}\n$$\n",
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+ "type": "text",
+ "text": "where the term $\\langle W - W_{i-1}, \\nabla \\ell(W_{i-1}; \\mathbf{k}_i, \\mathbf{v}_i) \\rangle$ is the local linear approximation of the original loss at time $i$ and the second term is a regularization term. While the first part $\\sum_{i=1}^{t} \\langle W - W_{i-1}, \\nabla \\ell(W_{i-1}; \\mathbf{k}_i, \\mathbf{v}_i) \\rangle$ measures how well can the memory learn all the past tokens, the second term $\\frac{1}{2\\eta} \\|W\\|_2^2$ penalizes the memory update with respect to the size of memory.",
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+ "text": "Equation (7) uses linear approximation of the loss function and quadratic regularization. We can, however, in principle use other approximations of the loss function as well as other regularization functions, as used in the past in online optimization (Hazan et al. 2016; Shalev-Shwartz et al. 2012) or in general optimization (Miral 2015; Razaviyayn et al. 2013). Such changes are the idea behind the development of other optimization algorithms such mirror descent. More specifically, we can generalize the update rule in (7) to the form:",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = \\arg \\min _ {W \\in \\mathcal {W}} \\underbrace {\\sum_ {i = 1} ^ {t} \\widehat {\\ell_ {i}} (W ; \\mathbf {k} _ {i} , \\mathbf {v} _ {i})} _ {\\text {A t t e n t i o n a l B i a s}} + \\underbrace {\\frac {1}{\\eta_ {t}} \\mathcal {R} _ {t} (W)} _ {\\text {M e m o r y S t a b i l i t y}}. \\tag {FTRLViewpoint}\n$$\n",
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+ "type": "page_number",
+ "text": "5",
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+ "type": "text",
+ "text": "In this update rule, the term $\\sum_{i=1}^{t} \\widehat{\\ell}_i(W; \\mathbf{k}_i, \\mathbf{v}_i)$ aims at memorizing the tokens at test time, while the term $\\mathcal{R}_t(W)$ regularizes the learning dynamics and take the size of the memory into account when updating it by a new incoming data. Choosing different loss functions $\\widehat{\\ell}_i(W; x_i)$ and the regularization term $\\frac{1}{\\eta_t} \\mathcal{R}_t(W)$ can lead to different algorithms such as (online) gradient descent or mirror descent. In this generalization, $\\eta_t$ to can be data-dependent. Moreover, we will allow imposing constraint $\\mathcal{W}$ on the choice $W$ .",
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+ "type": "text",
+ "text": "3.3 Viewpoint 2: Learning the Latest Token While Retaining Previous Information",
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+ "text": "Another way to interpret the update rule (5) is to view it as learning from the latest key-value pair $(\\mathbf{k}_i, \\mathbf{v}_i)$ (via using its gradient or surprise metric), while staying close to the previous state $W_{t-1}$ to retain the previously memorized tokens. Formally, (5) is equivalent to",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = \\arg \\min _ {W} \\left\\langle W - W _ {t - 1}, \\nabla \\ell (W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) \\right\\rangle + \\frac {1}{2 \\eta_ {t}} \\left\\| W - W _ {t - 1} \\right\\| _ {2} ^ {2}\n$$\n",
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+ "text": "The first term locally approximates $\\ell(W; \\mathbf{k}_t, \\mathbf{v}_t)$ around the previous state $W_{t-1}$ , while the last term regularizes deviations from $W_{t-1}$ . This form can generalize to",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = \\arg \\min _ {W \\in \\mathcal {W}} \\underbrace {\\widetilde {\\ell_ {t}} (W ; \\mathbf {k} _ {t} , \\mathbf {v} _ {t})} _ {\\text {A t t e n t i o n a l B i a s}} + \\underbrace {\\operatorname {R e t} _ {t} (W , W _ {t - 1})} _ {\\text {R e t e n t i o n}}, \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\text {(L e a r n i n g - R e t a i n i n g V i e w p o i n t)}\n$$\n",
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+ "type": "text",
+ "text": "where the term $\\widetilde{\\ell_t} (W;\\mathbf{k}_t,\\mathbf{v}_t)$ is an approximation of $\\ell (W;\\mathbf{k}_t,\\mathbf{v}_t)$ and minimizing it corresponds to Learning from the new concepts $(\\mathbf{k}_t,\\mathbf{v}_t)$ . The second term $\\mathrm{Ret}_t(W,W_{t - 1})$ regularizes the changes in $W$ to make the learning dynamics stable and to retain previously learned knowledge. This Retention function may have local and global components:",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\operatorname {R e t} _ {t} \\left(W, W _ {t - 1}\\right) = \\underbrace {\\frac {1}{\\eta_ {t}} \\mathrm {D} _ {t} \\left(W , W _ {t - 1}\\right)} _ {\\text {L o c a l R e t e n t i o n}} + \\underbrace {\\frac {1}{\\alpha_ {t}} \\mathrm {G} _ {t} \\left(W\\right)} _ {\\text {G l o b a l R e t e n t i o n}}.\n$$\n",
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+ "type": "text",
+ "text": "Here, the term $\\mathrm{D}_t(W, W_{t-1})$ , which is a premetric that controls the deviations from $W_{t-1}$ , aims at retaining previously learned knowledge. The coefficient $\\eta_t$ can be viewed as a meta in-context learning rate, where larger values of $\\eta_t$ leads to learning more from new concepts, while allowing higher forgetting of previously learned concepts. The second term is a global retention that controls the change of the memory with respect to its size. The special instances of the above viewpoint (e.g., without global retention, with implicit closed-form solution, and/or with limited memory structure) have been the motivation behind some of the recent studies such as Liu et al. (2024a).",
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+ "type": "text",
+ "text": "3.4 Further Discussions on the Two Viewpoints",
+ "text_level": 1,
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+ "type": "text",
+ "text": "The (FTRL Viewpoint) and (Learning-Retaining Viewpoint) are connected through the lens of online optimization. For example, as discussed above, by choosing linear approximation of the loss and quadratic regularization/retention, they can both cover online gradient descent update in (5) as a special case. One straightforward way to make the connection explicit is by defining the premetric $\\mathrm{D}_t(W;W^{\\prime})$ based on the previous loss functions and the regularization, as described in Proposition 3.2 below:",
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+ "text": "Proposition 3.2. Let $\\eta_t = \\eta$ and define $h_t(W) \\coloneqq \\sum_{i=1}^{t-1} \\widehat{\\ell}_i(W; \\mathbf{k}_i, \\mathbf{v}_i) + \\frac{1}{\\eta} R(W)$ . Assume $\\mathcal{W} = \\mathbb{R}^d$ and the function $h_t(W)$ is strictly convex in $W$ and let $\\mathcal{D}_h(\\cdot, \\cdot)$ be the Bregman divergence defined by function $h(\\cdot)$ , i.e., $\\mathcal{D}_h(W, W') = h(W) - h(W') - \\langle \\nabla h(W'), W - W' \\rangle$ . Set $Ret_t(W, W') = \\mathcal{D}_h(W, W')$ and $\\widetilde{\\ell}_t(W; x_t) = \\widehat{\\ell}_t(W; x_t)$ in (Learning-Retaining Viewpoint). Then, the update rule in (Learning-Retaining Viewpoint) is equivalent to the update rule in (FTRL Viewpoint).",
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+ "type": "text",
+ "text": "We provide the proof in Appendix B. The above proposition shows that (Learning-Retaining Viewpoint) can also explain the approaches obtained by (FTRL Viewpoint), under some mild assumptions. Hence, (Learning-Retaining Viewpoint) may be seen as a more general version. This is why we focus on this viewpoint in most of our derivations in the next sections.",
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+ "type": "page_number",
+ "text": "6",
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+ "text": "Remark 3. Given the above viewpoint, we can see that even by using additional global regularization there is no memory erasing or forgetting process (a common term in modern architectures (Behrouz et al. 2024c; Yang et al. 2024a)) but the model might decide to not retain the past state of the memory. Interestingly, this observation also matches the human memory process, where brain does not erase memories but they might become inaccessible due to retrieval failures (Robertson 2002). Therefore, instead of calling it a forget gate, later on, we use \"Retention Gate\" to refer to this term.",
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+ "text": "Remark 4. As we discuss in Section 4 and summarize in Table 1, most existing modern sequence models are optimizing associative memory objective (attentional bias in Equation 4) using gradient descent. Therefore, to provide further intuition about the connection of existing sequence models as well as their online learning interpretations, we discuss the above two viewpoints that are limited to gradient descent-based update rules. Our initial definition of attentional bias and associative memory in Equation 4, however, is broader and can be optimized by any optimization algorithm (e.g., even Newton's method, or non-parametric solutions).",
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+ "type": "text",
+ "text": "4 MirAs: Learning to Memorize with Robust and Expressive Memory",
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+ "text": "Building upon our definition of associative memory, attentional bias, and previous viewpoints, we present MIRAs framework that not only accurately unifies existing backbone architectures but it also provides insights on how to design the next generation of sequence models. As discussed earlier in Section 3, learning an associative memory can be interpreted as a meta-learning task, in which the associative memory learns how to compress and store data into its parameters at test time. The architecture of the memory in such tasks is particularly important as in longer contexts, the expressivity of the memory structure can limit its ability to learn the underlying patterns. Therefore, the first choice to design a sequence model is the structure of the memory. Given the structure of the memory, parameterized by a set of parameters $W$ , as discussed earlier, we aim to minimize a loss function $\\ell(W; \\cdot, \\cdot)$ with a retention regularizer $\\mathrm{Ret}(\\cdot)$ via a learning algorithm (e.g., gradient descent). Accordingly, MIRAs requires four design choices:",
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+ "1. Memory Structure: This choice specifies the architecture of the memory. For example, this architecture can be a vector, a linear function, a Multilayer Perceptron (MLP) layer, or even more complex structures. We may restrict the choice of $W$ to be within a certain region, e.g., $W$ to lie within an $L_{2}$ ball to avoid infinite values or unstable training.",
+ "2. Attentional Bias: A key choice is the attentional bias objective $\\mathcal{L}(\\cdot)$ in Equation 4. We can even consider different approximations of the loss function, (e.g., $\\widehat{\\ell} (\\cdot ,\\cdot)$ in (FTRL Viewpoint) or $\\widetilde{\\ell} (\\cdot ,\\cdot)$ in (Learning-Retaining Viewpoint)). The choice of attentional bias determines how memory memorizes the context, maps the inputs, and prioritizes the events.",
+ "3. Memory Stability and Retention: Another key choice is the retention regularizer $\\mathcal{R}(\\cdot)$ (e.g., $\\mathcal{R}_t(\\cdot)$ in (FTRL Viewpoint) and $\\mathrm{Ret}_t(\\cdot)$ in (Learning-Retaining Viewpoint)). In parametric setups, this choice balances learning with retention of past state. An effective retention gate is key to the good performance in long context tasks.",
+ "4. Memory Algorithm: Finally, this choice specifies the learning algorithm that we use to optimize the memory objective. One may use gradient descent, gradient descent with momentum, or any other algorithm (including finding non-parametric solutions)."
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+ "type": "text",
+ "text": "The above choices are major design choices for designing backbone sequence models in neural architectures. There are, however, minor decisions that can distinguish models; i.e., data-dependent or independent parameters, scalar or channel-wise learning rate/retaining gate, etc. Next, we discuss the overview of how existing architectures fit into MIRAS framework.",
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+ "type": "text",
+ "text": "RNNs with Hebbian Rule. The first generation of modern recurrent architectures (e.g., Linear attention (Katharopoulos et al. 2020), RetNet (Sun et al. 2023), Mamba (Gu et al. 2024), and GLA (Yang et al. 2024b)) are based on Hebbian-like (e.g., gated Hebbian) learning rule (Hebb 2005). We let attentional bias be the dot product similarity. That is, given a memory $\\mathcal{M} \\in \\mathbb{R}^{d \\times n}$ and $\\mathbf{k}, \\mathbf{v} \\in \\mathbb{R}^d$ , we define $\\tilde{\\ell}_t \\coloneqq -2\\langle \\mathcal{M}_t \\mathbf{k}_t, \\mathbf{v}_t \\rangle$ and local retention as $\\mathrm{Ret}_t(\\mathcal{M}, \\mathcal{M}_{t-1}) = \\| \\mathcal{M}_t - \\alpha \\mathcal{M}_{t-1} \\|_F^2$ . Using Equation Learning-Retaining Viewpoint and gradient descent as the optimizer (i.e., memory learning algorithm), the memory update rule is:",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {M} _ {t} = \\alpha \\mathcal {M} _ {t - 1} + \\mathbf {v} _ {t} \\mathbf {k} _ {t} ^ {\\top}. \\tag {8}\n$$\n",
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+ "text": "When (1) $\\alpha = 1$ , memory update is equivalent to Linear Attention (LA) (Katharopoulos et al. 2020); (2) $\\alpha \\in \\mathbb{R}$ is a learnable parameter, resulting architecture is either lightening attention ( $n > 1$ ) (Li et al. 2025) or RetNet ( $n = 1$ ) (Sun et al. 2023); and (3) $\\alpha_{t} \\in \\mathbb{R}$ are data-dependent learnable parameters, resulting sequence model is Mamba2 (Dao et al. 2024).",
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+ "type": "page_number",
+ "text": "7",
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+ "type": "text",
+ "text": "RNNs with Delta Rule. To improve the memory management and to enhance the memory capacity of the above group, several studies suggest using delta rule (Neil et al. 2017; Schlag et al. 2021) as the learning algorithm in recurrent neural networks (e.g., DeltaNet (Schlag et al. 2021), Longhorn (Liu et al. 2024a), and RWKV7 (Peng et al. 2025b)). In this part, we recall that where $\\mathcal{M} \\in \\mathbb{R}^{d \\times n}$ , delta rule is equivalent to optimizing MSE objective $\\| \\mathcal{M}_t \\mathbf{k}_t - \\mathbf{v}_t \\|_2^2$ with $\\mathrm{Ret}_t(\\mathcal{M}, \\mathcal{M}_{t-1}) = \\| \\mathcal{M}_t - \\alpha \\mathcal{M}_{t-1} \\|_F^2$ as local retention, and stochastic gradient descent as optimizer: ( $\\eta_t$ is defined in Equation Learning-Retaining Viewpoint)",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {M} _ {t} = \\alpha \\left(\\mathbf {I} - \\eta_ {t} \\mathbf {k} _ {t} \\mathbf {k} _ {t} ^ {\\top}\\right) \\mathcal {M} _ {t - 1} + \\mathbf {v} _ {t} \\mathbf {k} _ {t} ^ {\\top}. \\tag {9}\n$$\n",
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+ "text": "When (1) $\\alpha = 1$ , memory update is equivalent to DeltaNet (Schlag et al. 2021); and (2) $\\alpha_{t} \\in \\mathbb{R}^{m}$ are data-dependent learnable parameters, resulting sequence model is either Gated DeltaNet (Yang et al. 2024a) ( $m = 1$ ), or RWKV7 (Peng et al. 2025b) ( $m = d$ ). Therefore, RNNs with delta rule are special instances of MIRAS.",
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+ "text": "Beyond Delta Rule. As discussed earlier, while delta rule with its value replacement strategy is more powerful than Hebbian-like learning rules, it suffers from theoretical limitations (Irie et al. 2023) and achieves moderate performance in practice (Yang et al. 2024c). Therefore, several studies have focused on update rules beyond delta rule. Recently, Titans (Behrouz et al. 2024c) suggests using non-linear MSE objective of $\\| \\mathcal{M}_t(\\mathbf{k}_t) - \\mathbf{v}_t\\| _2^2$ with both local and global retention of $\\mathrm{D}_t = \\| W_t - W_{t - 1}\\| _F^2$ and $\\mathrm{G}_t = \\| W_t\\| _2^2$ and optimize it with gradient descent with momentum $^2$ . Therefore, Titans-LMM is a special instance of MIRAs, where we use the abovementioned attentional bias and retention regularizations, and gradient descent with momentum as the optimizer.",
+ "bbox": [
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+ "type": "text",
+ "text": "Another example of such models is Mesa-layer, in which the model uses $\\sum_{i=1}^{t} \\|\\mathcal{M}_{t}(\\mathbf{k}_{i}) - \\mathbf{v}_{i}\\|_{2}^{2}$ as the attentional bias objective with $\\|\\mathcal{M}_{t}\\|_{2}^{2}$ as the retention regularization. Since these models use Newton's method to optimize such an objective, they provide a more expressive update rule than delta rule. We further discuss a set of new learning algorithms beyond delta rule in Section 5.",
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+ "type": "text",
+ "text": "Attention. As discussed by Sun et al. (2024), softmax attention is a non-parametric solution of $\\ell_2$ -MSE loss function (i.e., $\\| W\\mathbf{k} - \\mathbf{v}\\| _2^2$ ) with Nadaraya-Watson estimator. Therefore, softmax attention is an instance of MIRAS, when we find the non-parametric solution to the MSE loss with Nadaraya-Watson estimator, without retention.",
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+ "type": "text",
+ "text": "5 Beyond Existing Attentional Biases and Retention Gates",
+ "text_level": 1,
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+ "text": "As discussed in the previous section, existing work focuses only on linear/quadratic choices for the attentional bias or retention gate. In particular, the loss function $L(\\mathcal{M}(\\mathbf{k}_t),\\mathbf{v}_t)$ is defined as $L(\\mathcal{M}(\\mathbf{k}_t),\\mathbf{v}_t) = c_t\\| \\mathcal{M}(\\mathbf{k}_t) - \\mathbf{v}_t\\|^2$ for some (learnable) constant $c_{t}$ in prior work. Also the regularization term $R_{t}(W)$ or the parametric $D_{t}$ is considered as a quadratic/linear function. In addition, almost all prior work considers $W$ to be the entire $\\mathbb{R}^d$ space. However, in general there could be various choices for all the three aforementioned design choices. To illustrate the potential and flexibility of our designed framework, here, we review some of the potential design choices for attentional bias and retention gate in MirAS. For the sake of clarity, we discuss all these attentional bias and memory retention gates based on using gradient descent as the optimizer, and so based on the provided two view points. However, these attentional bias objectives and retention regularizers can be directly used in Equation 4 and optimized by using any other optimization algorithms, resulting in different update rules.",
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+ "type": "text",
+ "text": "5.1 Alternative Attentional Biases",
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+ "text": "Variant 1: $\\ell_p$ -Attentional Bias. As discussed in the main body, attentional bias defines the \"similarity metric\" and measures how well memory can recall the value, given its corresponding key. Although $\\ell_2$ regression loss often is a natural choice, it is sensitive to noise in the data. A natural extension is to use $\\ell_p$ -norm class of objectives. That is, let $\\mathcal{M}$ be the memory, $\\mathbf{k}$ be the keys, and $\\mathbf{v}$ be the values, we define $\\ell_p$ -attentional bias as:",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {L} \\left(\\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right); \\mathbf {v} _ {t}\\right) = \\| \\mathcal {M} \\left(\\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {p} ^ {p}, \\tag {10}\n$$\n",
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+ "type": "page_footnote",
+ "text": "The retention gate (forget gate) in Titans is different from Mamba2 and Gated DeltaNet that we discussed above. The main difference comes from the case of full memory erase. While Mamba2 gating removes the entire memory and treats the next token as the first ever seen data, Titans use a \"cold start\" strategy and use the previous state of the memory to measure the surprise of the incoming token before fully erasing the memory.",
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+ "type": "page_number",
+ "text": "8",
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+ "type": "text",
+ "text": "where $p \\in \\mathbb{R}^{\\geq 1}$ and $\\| . \\|_p$ is the $p$ -norm. Although depending on the distribution of the data, we might want to use different values of $p$ (see Section 6), different values of $p$ can result in memory architectures with interesting properties. For the sake of simplicity, let memory be a matrix, i.e., $W \\in \\mathbb{R}^{m \\times d}$ and $\\mathcal{M}(W, \\mathbf{k}_t) = W\\mathbf{k}_t$ , the closed form can be derived as:",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = W _ {t} - \\eta_ {t} \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) = W _ {t} - p \\eta_ {t} \\left(\\operatorname {S i g n} \\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\odot | W \\mathbf {k} _ {t} - \\mathbf {v} _ {t} | ^ {p - 1}\\right) \\mathbf {k} _ {t} ^ {\\top}. \\tag {11}\n$$\n",
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+ "type": "text",
+ "text": "Let $p = 1$ , the recurrence is simplified as:",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = W _ {t} - \\eta_ {t} \\operatorname {S i g n} \\left(W _ {t} \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {\\top}, \\tag {12}\n$$\n",
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+ "type": "text",
+ "text": "which means that the memory has only two values of $-1$ and $1$ . We call this variation value-less associative memory, in which we store entities (keys) but map them into two extreme class of $-1$ and $+1$ .",
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+ "text": "Remark 5. One of the critical challenges to use the above update rule is in the backpropagation process, in which $\\operatorname{Sign}(\\cdot)$ and $|\\cdot|$ are non-differentiable and so might cause unstable training. To overcome this issue, we use $\\operatorname{Sign}(x) \\approx \\tanh(\\alpha x)$ , and $|x| = \\sqrt{x^2 + \\epsilon}$ , as the smooth approximators of these functions.",
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+ "type": "text",
+ "text": "One simple interpretation for such behavior (i.e., value-less memory) is similar to the coping mechanism in humans (Loftus 1993), in which the memory does not store the values for extreme events. This interpretation of protective memory in extreme events motivates our next variant.",
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+ "type": "text",
+ "text": "Variant 2: Huber Loss: Memory with Coping Mechanism. While $\\ell_2$ -norm objective is a common choice for many statistical and machine learning tasks, it is known to be sensitive to outliers and extreme samples. This sensitivity extends to the use of $\\ell_2$ loss for attentional bias. To address this and drawing motivation from robust regression literature, we suggest utilizing the Huber loss-type (Hastie et al. 2009; Huber 1992) as the attentional bias, thereby reducing the negative impact of the outlier data on the memory learning process.",
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+ "type": "text",
+ "text": "We can apply Huber-type loss in three different ways: The first approach is to define the summation of the Huber loss across different coordinates as the total loss, i.e.,",
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+ "type": "equation",
+ "text": "\n$$\n\\ell (W; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) = \\sum_ {j} \\mathcal {H} (\\mathcal {M} (W, \\mathbf {k} _ {t}) _ {j} - \\mathbf {v} _ {t, j}),\n$$\n",
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+ "type": "text",
+ "text": "where $\\mathcal{M}(W,\\mathbf{k}_t)_j$ and $\\mathbf{v}_{t,j}$ denote the $j$ -th coordinate of $\\mathcal{M}(W,\\mathbf{k}_t)$ and $\\mathbf{v}_t$ respectively. The function $\\mathcal{H}(\\cdot):\\mathbb{R}\\mapsto \\mathbb{R}$ is the Huber loss defined as",
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+ "text": "\n$$\n\\mathcal {H} (a) = \\left\\{ \\begin{array}{l l} \\frac {1}{2} a ^ {2} & \\text {i f} | a | \\leq \\delta \\\\ \\delta \\left(| a | - \\frac {1}{2} \\delta\\right) & \\text {i f} | a | > \\delta . \\end{array} \\right. \\tag {13}\n$$\n",
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+ "type": "text",
+ "text": "Utilizing this attentional bias can lead to various memory update rules. For example, for the matrix form memory $\\mathcal{M}(W,\\mathbf{k}_t) = W\\mathbf{k}_t$ , the update rule is given by",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = W _ {t - 1} - \\eta_ {t} \\left[ \\left(\\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {T}\\right) \\odot \\left(\\mathbf {I} \\left(\\left| W \\mathbf {k} _ {t} - \\mathbf {v} _ {t} \\right| \\leq \\delta_ {t}\\right) \\mathbf {1} ^ {\\top}\\right) + \\left(\\delta_ {t} \\operatorname {S i g n} \\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\mathbf {k} ^ {\\top}\\right) \\odot \\left(\\mathbf {I} \\left(\\left| W \\mathbf {k} _ {t} - \\mathbf {v} _ {t} \\right| > \\delta_ {t}\\right) \\mathbf {1} ^ {\\top}\\right) \\right] \\tag {14}\n$$\n",
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+ "type": "text",
+ "text": "In this formulation, the parameter $\\delta_t$ decides the type of the memory used for each block of memory ( $\\ell_2$ -norm objective or value-less) based on the context, making the memory more robust to outliers.",
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+ "type": "text",
+ "text": "The second approach is to define the Huber-type loss based on the $\\ell_2$ loss over all coordinates, i.e.,",
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+ "type": "equation",
+ "text": "\n$$\n\\ell (W; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) = \\mathcal {H} (\\| \\mathcal {M} (W, \\mathbf {k} _ {t}) - \\mathbf {v} _ {t} \\| _ {2}).\n$$\n",
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+ "type": "text",
+ "text": "For simplicity of derivations, assume matrix memory $M(W,\\mathbf{k}_t) = W\\mathbf{k}_t$ . Then using gradient descent for updating memory leads the memory update rule",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = W _ {t - 1} - \\eta_ {t} \\left\\{ \\begin{array}{l l} \\left(\\mathcal {M} \\left(W _ {t - 1}, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {T} & \\text {i f} \\| \\mathcal {M} \\left(W _ {t - 1}, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2} \\leq \\delta_ {t}, \\\\ \\delta_ {t} \\frac {\\left(\\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}\\right)}{\\| \\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2}} \\mathbf {k} _ {t} ^ {T} & \\text {O t h e r w i s e .} \\end{array} \\right. \\tag {15}\n$$\n",
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+ "type": "text",
+ "text": "Again, in the form (15), the parameter $\\delta_t$ decides the type of the memory used ( $\\ell_2$ -norm objective or normalized version) based on the context, making the memory more robust to outliers.",
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+ "type": "page_number",
+ "text": "9",
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+ "type": "text",
+ "text": "Finally, in the third approach, we present a smooth mixture method, in which the memory decides if for an incoming data it is better to use $\\ell_2$ or $\\ell_1$ attentional bias:",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = W _ {t - 1} - \\left\\{ \\begin{array}{l l} \\eta_ {t} \\nabla \\ell_ {2} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) & \\text {i f} \\| \\mathcal {M} (\\mathbf {k} _ {t}) - \\mathbf {v} _ {t} \\| \\leq \\delta_ {t}, \\\\ \\eta_ {t} \\delta_ {t} \\nabla \\ell_ {1} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) & \\text {O t h e r w i s e .} \\end{array} \\right. \\tag {16}\n$$\n",
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+ "type": "text",
+ "text": "The role of parameter $\\delta_t$ is the same as above.",
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+ "text": "Variant 3: Memory Robust to Value Shifts. Following the robustness requirement discussed in the previous section, we aim to design a memory mechanism that exhibits resilience against small shifts in the value parameter. A natural approach in this context is to employ a robust optimization formulation. Specifically, we define the loss function as the worst-case $\\ell_2$ distance between the predicted memory output and the perturbed true value:",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {L} \\left(\\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right); \\mathbf {v} _ {t}\\right) = \\max _ {\\| \\delta \\mathbf {v} _ {t} \\| _ {2} \\leq \\Delta} \\frac {1}{2} \\| \\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right) - \\left(\\mathbf {v} _ {t} + \\boldsymbol {\\delta} \\mathbf {v} _ {t}\\right) \\| _ {2} ^ {2}. \\tag {17}\n$$\n",
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+ "type": "text",
+ "text": "This formulation seeks the memory parameters $W$ that perform well even under the adverse local perturbation of the true value $\\mathbf{v}_t$ within an $\\ell_2$ ball of radius $\\Delta$ . To solve the maximization problem in (17), we find the optimal perturbation $\\delta \\mathbf{v}_t^*$ . By solving this problem with respect to $\\delta \\mathbf{v}_t$ , we arrive at:",
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+ "text": "\n$$\n\\delta \\mathbf {v} _ {t} ^ {*} = \\Delta \\frac {- \\mathcal {M} (W , \\mathbf {k} _ {t}) + \\mathbf {v} _ {t}}{\\| \\mathcal {M} (W , \\mathbf {k} _ {t}) - \\mathbf {v} _ {t} \\| _ {2}}\n$$\n",
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+ "text": "Substituting this optimal perturbation back into the loss function (17), we obtain the robust loss:",
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+ "text": "\n$$\n\\mathcal {L} \\left(\\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right); \\mathbf {v} _ {t}\\right) = \\frac {1}{2} \\| \\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2} ^ {2} + \\Delta \\| \\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2} + \\frac {1}{2} \\Delta^ {2}.\n$$\n",
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+ "type": "text",
+ "text": "This robust loss function is a combination of the standard $\\ell_2$ loss and a term proportional to the $\\ell_2$ norm of the error, scaled by the robustness parameter $\\Delta$ . The value of $\\Delta$ thus controls the trade-off between fitting the nominal data and ensuring robustness against value perturbations.",
+ "bbox": [
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+ "type": "text",
+ "text": "For simplicity of the derivations, let us consider a constant value for $\\Delta$ , an Euclidean retention gate $\\mathrm{Ret}_t(W,W_{t - 1}) = \\| W - W_{t - 1}\\|^2$ , and an attentional bias term $\\widetilde{\\ell} (W;\\mathbf{k}_t,\\mathbf{v}_t) = \\langle W - W_{t - 1},\\nabla \\ell (W_{t - 1};\\mathbf{k}_t,\\mathbf{v}_t)\\rangle$ . Furthermore, to simplify the memory operation, we assume a linear matrix memory model $\\mathcal{M}(W,\\mathbf{k}_t) = W\\mathbf{k}_t$ . Under these assumptions, we can derive the memory update mechanism using gradient descent on the robust loss:",
+ "bbox": [
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = W _ {t - 1} - \\eta \\left(\\left(\\mathcal {M} \\left(W _ {t - 1}, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {\\top} + \\Delta \\frac {\\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}}{\\| \\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2}} \\mathbf {k} _ {t} ^ {\\top}\\right)\n$$\n",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "In this update rule, the parameter $\\Delta$ , which governs the influence of the robustness term, can also be treated as a learnable parameter, allowing the model to adapt its robustness based on the observed data.",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "5.2 Alternative Retention Gates",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Variant 1: Memorization Over A Scaled Probability Simplex Via $f$ -Divergence. A common technique in learning to prevent numerical instabilities and exploding values is to restrict the search space to a bounded domain. Following this principle, to avoid numerical instabilities, we can constrained the variable $W_{t}$ to lie within a (scaled) probability simplex. In other words, we can restrict the state to lie in the constraint set",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {W} = \\{W \\mid \\| W \\| _ {1} = c \\text {a n d} W _ {j l} \\geq 0, \\forall j, l \\}.\n$$\n",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "In this set, each matrix $W$ can be viewed as a measure. Thus, in (Learning-Retaining Viewpoint), we can utilize divergences over measures to define our premetric. For example, we can use $f$ -divergence measure (Polyanskiy et al. 2025, Def 4.9), (Csiszar 1967) to define $\\mathrm{D}_t(\\cdot, \\cdot)$ . More specifically, let $f(\\cdot)$ be a smooth strictly convex function from $\\mathbb{R}^+$ to $\\mathbb{R}$ with $f(1) = 0$ . Then, we can define the $f$ -divergence between $W$ and $W'$ as",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathrm {D} _ {t} (W, W ^ {\\prime}) = \\sum_ {j l} W _ {j l} ^ {\\prime} f \\left(\\frac {W _ {j l}}{W _ {j l} ^ {\\prime}}\\right).\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 9
+ },
+ {
+ "type": "page_number",
+ "text": "10",
+ "bbox": [
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+ "type": "text",
+ "text": "It is known that $f$ -divergence is zero if and only if $W = W'$ ; see Polyanskiy et al. 2025, Theorem 2.3. Using the above premetric as the retention gate and setting $\\widetilde{\\ell}(W; \\mathbf{k}_t, \\mathbf{v}_t) = \\langle W - W_{t-1}, \\nabla \\ell(W_{t-1}; \\mathbf{k}_t, \\mathbf{v}_t) \\rangle$ in (Learning-Retaining Viewpoint), we get the update rule",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nW _ {t} = W _ {t - 1} \\odot g \\left(- \\zeta_ {t} - \\eta_ {t} \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right). \\tag {18}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Here $g(\\cdot)$ is the inverse of the mapping $f'$ , i.e., $g(f'(\\tau)) = \\tau$ , $\\forall \\tau$ ; the operator $\\odot$ denotes the Hadamard (elementwise) product, and $\\zeta_t$ should be chosen such that $\\| W_t\\|_1 = c$ . Notice that since the function $f(\\cdot)$ is strictly convex and smooth, its derivative is strictly increasing and hence $g(\\cdot)$ is well defined. Conversely, for any strictly monotone function $g(\\cdot)$ , we can find its inverse function $g^{-1}$ (which is strictly increasing) and define $f(\\tau) = \\mathrm{const} + \\int_{\\tau' = 0}^{\\infty}g^{-1}(\\tau')d\\tau'$ . The term const should be chosen such that $f(1) = 0$ . Then the update rule in (18) can be interpreted by the $f$ -divergence regularization, as explained above. Therefore, one can directly choose a continuous monotonically increasing function $g(\\cdot)$ and use (18) for memory update.",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Specializing to KL divergence. Let us further make the above update rule explicit by using special function $f$ . If we choose $f(\\tau) = \\tau \\ln(\\tau)$ , then the $f$ -divergence becomes the widely used KL divergence measure $D_t(W, W_{t-1}) = \\sum_{jl} W_{jl} \\log \\left( \\frac{W_{jl}}{(W_t)_{jl}} \\right)$ . In addition, we can also utilize the Shannon entropy as the global retention by regularizing deviations from uniform distribution, i.e., $G_t(W) = \\sum_{jl} W_{jl} \\log (W_{jl})$ . Combining these choices of the local and global retention gates, we obtain the overall retention gate",
+ "bbox": [
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\operatorname {R e t} _ {t} (W, W _ {t - 1}) = \\frac {1}{\\eta_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(\\frac {W _ {j l}}{\\left(W _ {t}\\right) _ {j l}}\\right) + \\frac {1}{\\alpha_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(W _ {j l}\\right)\n$$\n",
+ "text_format": "latex",
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+ {
+ "type": "text",
+ "text": "Choosing the attentional bias $\\widetilde{\\ell}(W; \\mathbf{k}_t, \\mathbf{v}_t) = \\langle W - W_{t-1}, \\nabla \\ell(W_{t-1}; \\mathbf{k}_t, \\mathbf{v}_t) \\rangle$ and the above retention gate will lead to the update rule",
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+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nW _ {t} = \\arg \\min _ {W} \\left\\langle W - W _ {t - 1}, \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) \\right\\rangle + \\frac {1}{\\eta_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(\\frac {W _ {j l}}{\\left(W _ {t}\\right) _ {j l}}\\right) + \\frac {1}{\\alpha_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(W _ {j l}\\right) \\tag {19}\n$$\n",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\text {s . t .} \\quad \\sum_ {j l} W _ {j l} = c, W _ {j l} \\geq 0, \\forall j l \\tag {20}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 10
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+ {
+ "type": "text",
+ "text": "Attaching the Lagrange multiplier to the first constraint, the KKT conditions imply",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left(\\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right) _ {j l} + \\left(\\frac {1}{\\eta_ {t}} + \\frac {1}{\\alpha_ {t}}\\right) \\left(1 + \\log W _ {j l}\\right) - \\frac {1}{\\eta_ {t}} \\log \\left(\\left(W _ {t - 1}\\right) _ {j l}\\right) + \\mu_ {t} = 0, \\quad \\forall j, l\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "where $\\mu_t$ should be chosen such that $\\sum_{jl} W_{jl} = c$ . Rearranging the terms and defining $\\lambda_t = \\frac{1 / \\alpha_t}{1 / \\alpha_t + 1 / \\eta_t}$ , $\\eta_t' = \\frac{1}{1 / \\alpha_t + 1 / \\eta_t}$ , we get the update rule",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nW _ {t} \\leftarrow c \\operatorname {S o f t m a x} \\left(\\left(1 - \\lambda_ {t}\\right) \\log \\left(W _ {t - 1}\\right) - \\eta_ {t} ^ {\\prime} \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right) \\tag {21}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "where $\\lambda_t \\in (0,1)$ and $\\eta' \\in \\mathbb{R}^+$ are the parameters that can be learned during training. The Softmax operator ensures that the output lies in the set $\\mathcal{W}$ .",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Notice that while all above calculations are done for a matrix $W$ , similar update rule holds for other forms of parameters such as when $W$ is a neural network (or when the parameter $W$ is normalized per slice).",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Variant 2: Elastic Net Regularization: Hard and Soft Forgetting. Elastic net is a powerful and popular tool in regression analysis to balance the feature selection capabilities of LASSO (Tibshirani 1996) and bias reduction properties of Ridge regression (Hilt et al. 1977; Hoerl et al. 1970). It has been widely used in different applications due to its ability to handle high-dimensional data and mitigate the effects of multicollinearity. Given this success, a natural question is what happens if we use this regularization scheme in our context.",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Let us start based on (Learning-Retaining Viewpoint) to design our memorization scheme. In (Learning-Retaining Viewpoint), we discussed that the loss function $\\widetilde{\\ell_t} (W;\\mathbf{k}_t,\\mathbf{v}_t)$ is an approximation of the original function $\\ell (\\cdot)$ , measuring our goodness-of-fit. Regularizing this loss with elastic net regularizer, we obtain the approximation",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\widetilde {\\ell} _ {t} (W; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) = \\langle W - W _ {t - 1}, \\nabla \\ell (W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) \\rangle .\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "page_number",
+ "text": "11",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "with a global retention of $\\mathrm{G}_t(W) = \\frac{1}{2\\beta} \\| W\\| _2^2 +\\frac{1}{\\alpha}\\| W\\| _1$ . To fully specify the update rule of (Learning-Retaining Viewpoint), we also need to specify the premetric functions $\\mathrm{D}_t(\\cdot ,\\cdot)$ . For the sake of keeping the update rule simple (and parallelizable), we can choose",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathrm {D} _ {t} (W, W _ {t - 1}) = \\frac {1}{2} \\| W - W _ {t - 1} \\| _ {2} ^ {2}.\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "These choices of the attentional bias and retention gate leads to the following update rule:",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nW _ {t} = \\mathcal {S} _ {Y} \\left(\\lambda W _ {t - 1} - \\zeta \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right), \\tag {22}\n$$\n",
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+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "where $\\gamma = \\frac{\\eta\\beta}{\\alpha(\\eta + \\beta)}$ , $\\lambda = \\frac{\\beta}{\\beta + \\eta}$ , $\\zeta = \\eta\\lambda$ , and $S_{\\gamma}$ is the soft thresholding operator, applied element-wise. For each element, this operator is defined as",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathcal {S} _ {\\gamma} (z) = \\operatorname {s i g n} (z) \\max \\left\\{0, | z | - \\gamma \\right\\}.\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "text",
+ "text": "In other words, for large values of $z$ , $S_{\\gamma}(z)$ makes $z$ closer to zero by $\\gamma$ amount. If it is already in the $\\gamma$ -vicinity of zero, then it makes it zero (hard forget).",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Equation (22) can be viewed as a combination of soft forgetting (obtained by multiplying $W$ by $\\lambda \\in (0,1)$ , and a hard forgetting (if it is smaller than $\\gamma$ ). The hyperparameters $\\gamma, \\lambda,$ and $\\zeta$ can be learned. Notice that since the shrinkage operator is not differentiable, we can approximate it with its smooth approximation. For example, we can use $S_{\\gamma}(z) \\approx \\frac{|z|*\\arctan(z / \\gamma)}{\\pi / 2}$ .",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Variant 3: Elastic Net Regularization: Forgetting via Soft-thresholding. The elastic net regularizer can also be used in the (FTRL Viewpoint). In particular, in (FTRL Viewpoint), we can set",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\frac {1}{\\eta_ {t}} R _ {t} (W) = \\frac {1}{\\eta} \\| W \\| ^ {2} + \\frac {1}{\\alpha} \\| W \\| _ {1}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "and use $\\widehat{\\ell}(W; x_i) = \\langle W - W_{i-1}, \\nabla \\ell(W_{i-1}; x_i) \\rangle$ . Assuming initialization at $W_0 = 0$ , these choices of attentional bias and retention gate leads to the update rules:",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nA _ {t} = A _ {t - 1} - \\eta \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nW _ {t} = \\mathcal {S} _ {\\eta / \\alpha} (A _ {t}) \\tag {23}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Here $S_{\\eta /\\alpha}(\\cdot)$ is the soft-thresholding operator with parameter $\\eta /\\alpha$ , which can be smoothly as explained in Variant 1.1.",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Variant 4: General $L_{q}$ Memory Stability. Existing work is based on the retention gate choices $\\mathrm{D}_t(W, W_{t-1}) = \\|W - W_{t-1}\\|_F^2$ or $R(W) = \\|W\\|_2^2$ . However, one can choose other choices of retention gate. For example, in (FTRL Viewpoint), we can choose $L_{q}$ norm as the regularizer $R(W)$ . More specifically, for $1 < q \\leq 2$ , we can set",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\frac {1}{\\eta_ {t}} R (W) = \\frac {1}{2 \\eta (q - 1)} \\| W \\| _ {q} ^ {2}.\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Using this retention gate and choosing $\\widehat{\\ell_i} (W;\\mathbf{k}_t,\\mathbf{v}_t) = \\langle W - W_{i - 1},\\nabla \\ell (W_{i - 1};\\mathbf{k}_t,\\mathbf{v}_t)\\rangle$ in (FTRL Viewpoint), leads to the update rule $W_{t} = -\\eta \\frac{A_{t}}{\\|A_{t}\\|_{p}^{p - 2}}$ , where $p = \\frac{q}{q - 1}$ and $A_{t} = \\sum_{i = 1}^{t}\\nabla \\ell (W_{i - 1};\\mathbf{k}_{t},\\mathbf{v}_{t})$ ; see Shalev-Shwartz et al. 2012, Section 2.6. Here, $\\odot$ denotes the Hadamard (element-wise) product and $|\\cdot |$ is the element-wise absolute value operator. Assuming $W_0 = 0$ , this update rule can be recursively written as:",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nA _ {t} = A _ {t - 1} - \\eta \\nabla \\ell \\left(W _ {i - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right), \\quad \\text {a n d} \\quad W _ {t} = \\frac {A _ {t}}{\\| A _ {t} \\| _ {p} ^ {p - 2}}.\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Variant 5: Bregman Divergence as Retention Gate.. Another natural choice is to use Bregman divergence as retention gate, leading to a mirror descent-type algorithms. In particular, given a smooth strictly convex function $f(\\cdot): \\mathbb{R} \\mapsto \\mathbb{R}$ , we can define the function $F(W) = \\sum_{jl} f(W_{jl})$ . Based on this choice of function $F$ , we define the Bregman divergence",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nD _ {t} (W, W ^ {\\prime}) = F (W) - F \\left(W ^ {\\prime}\\right) - \\langle W ^ {\\prime}, W - W ^ {\\prime} \\rangle\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "page_number",
+ "text": "12",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "as our parametric function. Utilizing this retention gate and choosing $\\widetilde{\\ell}_t(W;\\mathbf{k}_t,\\mathbf{v}_t) = \\langle W - W_{t - 1},\\nabla \\ell (W_{t - 1};\\mathbf{k}_t,\\mathbf{v}_t)\\rangle$ in (Learning-Retaining Viewpoint), we obtain the update rule",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nW _ {t} = g \\left(- \\eta \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) + F ^ {\\prime} \\left(W _ {t - 1}\\right)\\right).\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "Here, $F'$ is the mapping obtained by applying $f'(\\cdot)$ (the derivative of $f$ ) element-wise to all entries of its input matrix argument. The function $g$ is the inverse of the mapping $F'(\\cdot)$ , i.e., $g(F'(W)) = W$ .",
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+ "type": "text",
+ "text": "If we choose $f(\\tau) = \\frac{\\tau^2}{2}$ , then $F'(W)$ becomes the identity mapping and so is $g$ . Therefore, the above update becomes simple gradient descent with no nonlinearity involved in the update rule. However, other choices of $f(\\cdot)$ introduces additional nonlinearity in $g(\\cdot)$ , which can enhance the expressivity of our memory. For example, we can choose the function $f(\\cdot)$ so that its derivative becomes the inverse sigmoid function, i.e., $f'(\\tau) = \\ln \\left( \\frac{\\tau}{1 - \\tau} \\right)$ with $f': (0,1) \\mapsto \\mathbb{R}$ . Since $f'(\\cdot)$ is strictly increasing, then the function $f(\\cdot)$ (and hence $F(\\cdot)$ ) is strictly convex. Therefore, the Bregman divergence is well defined. Moreover, the inverse of the function $f'(\\cdot)$ becomes the sigmoid function, i.e., $g(\\tau) = \\sigma(\\tau) = \\frac{\\exp(\\tau)}{1 + \\exp(\\tau)}$ with $g: \\mathbb{R} \\mapsto (0,1)$ . Then, the update of the memory becomes",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = \\sigma \\left(\\ln \\left(\\frac {W _ {t}}{1 - W _ {t}}\\right) - \\eta \\nabla \\ell (W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t})\\right),\n$$\n",
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+ "text": "where $\\sigma$ is the sigmoid function operated element-wise on the entries of $W$ , and the division operator $\\frac{W_t}{1 - W_t}$ is also performed element-wise. This update rule guarantees that the elements of $W_t$ remain within the interval $(0, 1)$ .",
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+ "type": "text",
+ "text": "5.3 MIRAs's Variants: MONETA, YAAD, and MEMORA",
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+ "text": "In the previous section we discussed different potential choices for attentional bias and retention gate to show the generality and the potential of MIRAs. In this section, building upon our framework, we present three novel sequence models, each of which designed based on a different motivation, and discuss how they can leverage fast parallel training.",
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+ "text": "MOnETA. Given $p,q\\in \\mathbb{R}^{\\geq 1}$ , we design $(p,q)$ -MONETA as the variant of MIRAs as follows: (1) For the choice of memory architecture, we use an MLP with 2 layers with expansion factor of 4 and GELU activation function (Hendrycks et al. 2016). We also use residual connections and layer norm, resulting in $\\mathcal{M}(x) = x + \\mathsf{LN}(W_1\\sigma (W_2x))$ . (2) We choose $\\ell_p$ -attentional bias (introduced in Equation 11) for MONETA. (3) For the choice of retention gate, we use the hybrid of $\\ell_q$ retention gate $\\frac{1}{2(q - 1)}\\| W\\| _q^2$ (see Section 5.2 for details) and the standard $\\ell_2$ regularization $\\frac{1}{\\beta}\\| W\\| _2^2$ . (4) Finally, we use gradient descent as the memory learning algorithm. The above choices, result in the following recurrent formula for the memory module:",
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+ "type": "equation",
+ "text": "\n$$\nA _ {t} = \\alpha_ {t} A _ {t - 1} - \\eta_ {t} \\nabla \\ell_ {p} \\left(W _ {i - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right), \\quad \\text {a n d} \\quad W _ {t} = \\frac {A _ {t}}{\\| A _ {t} \\| _ {q} ^ {q - 2}}. \\tag {24}\n$$\n",
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+ "text": "Notably the gradient can be calculated using:",
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+ "type": "equation",
+ "text": "\n$$\n\\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) = p \\eta_ {t} \\left(\\operatorname {S i g n} \\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\odot | W \\mathbf {k} _ {t} - \\mathbf {v} _ {t} | ^ {p - 1}\\right) \\mathbf {k} _ {t} ^ {\\top}. \\tag {25}\n$$\n",
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+ "text": "We use $(p,q) = (3,4)$ .",
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+ "text": "YAAD. Building upon our discussion on the importance of robust memory that protects itself from extreme events (tokens), we design YAAD based on Huber objective. That is, in MirAS, for the choice of memory structure, we follow MONETA and use an MLP with the same architecture as above; for the choice of attentional bias, we use Huber loss (defined in Equation 16); for the choice retention gate, for the sake of simplicity, we use a combination of local and global retention as $\\mathrm{Ret}_t(W,W_{t - 1}) = \\frac{1}{2\\theta_t}\\| W - W_{t - 1}\\| _F^2 +\\frac{1}{\\beta_t}\\| W\\| _2^2$ , which is equivalent to the \"forget gate\" mechanism introduced by Behrouz et al. (2024c); and finally, we simply use gradient descent as the memory learning algorithm. Given the above choices, we can write the resulted memory learning process as follows:",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = \\alpha_ {t} W _ {t - 1} - \\left\\{ \\begin{array}{l l} \\eta_ {t} \\nabla \\ell_ {2} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) & \\text {i f} \\| \\mathcal {M} (\\mathbf {k} _ {t}) - \\mathbf {v} _ {t} \\| \\leq \\delta_ {t}, \\\\ \\eta_ {t} \\delta_ {t} \\nabla \\ell_ {1} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) & \\text {O t h e r w i s e .} \\end{array} \\right. \\tag {26}\n$$\n",
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+ "text": "Note that for improving the expressive power, in all architectures, we decouple the learning rate $\\eta$ and the retention gate rate $\\alpha$ , resulting in an independent parameter $\\beta_{t} \\in [0,1]^{d}$ .",
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+ "text": "13",
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+ "img_path": "images/9cb01a969297d8878bd8358e093a6abd23c24cfb85585b92f9cd441c4a9e7943.jpg",
+ "image_caption": [
+ "Figure 2: Visualization of the MirAs's variant architecture, their hybrid counterpart with SWA, and block design of MirAs layer."
+ ],
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+ "text": "MEMORA. Finally, in MEMORA, we use the idea of elastic net regularization (i.e., hard and soft retention). To this end, in Miras: (1) For the choice of memory architecture, similar to above variants, we use an MLP (the same architecture as the previous variants). (2) For the choice of attentional bias, we use simple $\\ell_2$ regression loss. (3) For the choice of retention gate we use KL divergence as in Equation 21. (4) Finally, we optimize the memory using gradient descent, resulting in the following update rule:",
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+ "type": "equation",
+ "text": "\n$$\nW _ {t} = \\operatorname {S o f t m a x} \\left(\\alpha_ {t} \\log \\left(W _ {t - 1}\\right) - \\eta_ {t} \\nabla \\ell_ {2} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right) \\tag {27}\n$$\n",
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+ "text": "5.4 Architecture Backbone and Fast Training",
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+ "text": "Architectural Backbone. For the architectural backbone, we fully follow recent studies (Behrouz et al. 2024c; Yang et al. 2024a): We replace attention modules with our variants of MIRAs in Llama's macro architecture with MLPs with SwiGLU(. ) activation, rotary positional encodings (RoPE) (Su et al. 2024), and RMSNorm (Zhang et al. 2019). For MIRAs layer block, we follow the recent modern linear recurrent models (Behrouz et al. 2024c; Yang et al. 2024a), and incorporate a 1D depthwise-separable convolution layer (with kernel size of 4) after each of the query, key, and value projections. For the sake of training stability, we also use $\\ell_2$ normalization to $\\mathbf{q}$ and $\\mathbf{k}$ . The output of MIRAs layer block is normalized and gated with a linear layer (Mehta et al. 2023).",
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+ "text": "Channel-wise Parameters. For learnable parameters of $\\eta_t, \\delta_t$ and the retention gate of $\\alpha_t$ we use channel-wise parametrization, i.e., $\\eta_t, \\delta_t, \\alpha_t \\in \\mathbb{R}^d$ . While gaining more expressive power, this parametrization results in significant parameter increase. To mitigate this issue, following Peng et al. (2025b), we use low-rank projections to project the input into $\\mathbb{R}^k$ and then to $\\mathbb{R}^d$ , where $k$ is a hyperparameter (usually 32 or 64). The backbone architecture is illustrated in Figure 2.",
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+ "type": "text",
+ "text": "Hybrid Models. We also evaluate the hybrid version of Miras's variants. For hybrid models, we follow the Samba (Ren et al. 2024) architecture, in which we sequentially combine our Miras layer with Sliding Window Attention (SWA). The illustration of hybrid model Figure 2.",
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+ "type": "text",
+ "text": "Parallelizable Training. While the design of Miras's variant are theoretically well-motivated, their recurrence is non-linear, potentially making their straightforward training slow for large scales. In this section, we build upon the work of Behrouz et al. (2024c) and Sun et al. (2024) to make the training parallelizable. The main idea is to divide the sequence into",
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+ "type": "page_number",
+ "text": "14",
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+ "type": "text",
+ "text": "chunks with size $b$ (usually is 16 or 64) and calculate the gradient for all tokens in the current chunk with respect to the last state of the memory in the previous chunk. That is, we use $\\nabla \\ell(\\mathcal{M}_{t'}; \\mathbf{k}_t, \\mathbf{v}_t)$ instead of $\\nabla \\ell(\\mathcal{M}_{t-1}; \\mathbf{k}_t, \\mathbf{v}_t)$ , where $t'$ is the last state in the previous chunk.",
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+ "type": "text",
+ "text": "Given the above trick, we can calculate all gradients at once and make the recurrence inside each chunk linear. However, to fully take advantage of accelerators, we need to reformulate the process as matrix multiplication. For MONETA, for the sake of clarity, assume $q = 2$ . We follow the same algorithm as Behrouz et al. (2024c) and expand the recurrence as follows:",
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+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\mathcal {M} _ {t} = \\alpha_ {t} \\mathcal {M} _ {t - 1} - \\eta_ {t} \\nabla \\ell (\\mathcal {M} _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) \\\\ = \\beta_ {t} \\mathcal {M} _ {0} - \\sum_ {i = 1} ^ {t} \\eta_ {i} \\frac {\\beta_ {t}}{\\beta_ {i}} \\nabla \\ell \\left(\\mathcal {M} _ {t ^ {\\prime}}; \\mathbf {k} _ {i}, \\mathbf {v} _ {i}\\right), \\tag {28} \\\\ \\end{array}\n$$\n",
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+ "type": "text",
+ "text": "where $t' = t - \\mathrm{mod}(t, b)$ , and $\\beta_{i} = \\prod_{j=1}^{i} \\alpha_{j}$ . For the sake of clarity, we focus on the first chunk, i.e., $t = b$ and so $t' = 0$ , and explain the process for the case that $\\mathcal{M}_t = W_t$ is linear. The process for 2-layer MLPs and other chunks is similar. Using $\\ell_p$ loss function, we have:",
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+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\nabla \\ell \\left(W _ {0}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) = p \\left(\\operatorname {S i g n} \\left(W _ {0} \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\odot \\left| W _ {0} \\mathbf {k} _ {t} - \\mathbf {v} _ {t} \\right| ^ {p - 1}\\right) \\mathbf {k} _ {t} ^ {\\top} \\\\ \\Rightarrow \\sum_ {i = 1} ^ {b} \\eta_ {i} \\frac {\\beta_ {b}}{\\beta_ {i}} \\nabla \\ell \\left(W _ {0};; \\mathbf {k} _ {i}, \\mathbf {v} _ {i}\\right) = p \\mathbf {E} _ {b} \\odot \\mathbf {B} _ {b} \\odot \\operatorname {S i g n} \\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\odot \\left(\\left| W _ {0} \\mathbf {K} - \\mathbf {V} \\right| ^ {p - 1}\\right) \\mathbf {K} ^ {\\top}, \\tag {29} \\\\ \\end{array}\n$$\n",
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+ {
+ "type": "text",
+ "text": "where $\\mathbf{E}_b = \\left[\\eta_1\\quad \\eta_2\\quad \\dots \\quad \\eta_b\\right]$ and $\\mathbf{B}_b$ is defined analogously on $\\frac{\\beta_b}{\\beta_i}\\mathrm{s}$ . For the sake of stability in training, we use $\\operatorname{Sign}(x)\\approx \\tanh (\\alpha x)$ and $|x| = \\sqrt{x^2 + \\epsilon}$ , where $\\epsilon >0$ is a small number (i.e., $\\epsilon = 1e - 6$ ). As discussed in Equation 24, the case that $q\\neq 2$ appears as a normalization term on the memory. Similar to Titans (Behrouz et al. 2024c) and TTT (Sun et al. 2024), we do not apply this non-linearity inside each chunk and instead use it at the end of each chunk.",
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+ "type": "text",
+ "text": "For YAAD, the process is very similar to the above. We calculate the gradient of both $\\ell_1$ and $\\ell_2$ loss and use a masking based on $\\| \\mathcal{M}(\\mathbf{k}_t) - \\mathbf{v}_t\\| \\leq \\delta_t$ .",
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+ "type": "text",
+ "text": "For MEMORA, the update rule has two non-linear parts, i.e., softmax and log, making the model hardly parallelizable. To this end, as discussed above, we use its linear version inside each chunk and its non-linear version across chunks. However, using both log and softmax at the end of each chunk removes the effect of log. To this end, we consider a lag tokens after each chunk (i.e., tokens with index $i = kb + 1$ , where $b$ is the chunk size and $k \\in \\mathbb{Z}^+$ ). That is, let $\\mathcal{M}_0$ be the last state of the memory in previous chunk, we have:",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {M} _ {1} = \\operatorname {S o f t m a x} \\left(\\alpha_ {1} \\log \\left(\\mathcal {M} _ {0}\\right) - \\eta_ {1} \\nabla \\ell_ {2} \\left(\\mathcal {M} _ {0}; \\mathbf {k} _ {1}, \\mathbf {v} _ {1}\\right)\\right), \\tag {30}\n$$\n",
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+ "type": "text",
+ "text": "and then we use $\\mathcal{M}_1$ for the next chunk. Again, for the sake of clarity, assume that memory is linear, i.e., $\\mathcal{M}_1 = W_1$ :",
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+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\nabla \\ell \\left(W _ {1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) = \\left(W _ {1} \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {\\top} (31) \\\\ \\Rightarrow \\sum_ {i = 1} ^ {b} \\eta_ {i} \\frac {\\beta_ {b}}{\\beta_ {i}} \\nabla \\ell \\left(W _ {1};; \\mathbf {k} _ {i}, \\mathbf {v} _ {i}\\right) = \\mathbf {E} _ {b} \\odot \\mathbf {B} _ {b} \\odot \\left(W _ {1} \\mathbf {K} - \\mathbf {V}\\right) \\mathbf {K} ^ {\\top}, (32) \\\\ \\end{array}\n$$\n",
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+ "type": "text",
+ "text": "where matrices are defined the same as for Equation 29.",
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+ "type": "text",
+ "text": "6 Experiments",
+ "text_level": 1,
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+ "type": "text",
+ "text": "In our experimental evaluations, we aim to answer three main questions: (1) Does different attentional biases results in different architectures in practice? (2) How does different types of retention gates (i.e., retention gate) affect the performance of the model in long context? (3) How do MEMORA, MONETA, and YAAD perform in downstream tasks compare to baselines?",
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+ "type": "text",
+ "text": "Setup. We train our models with training context window of size 4096 using either FineWeb-Edu dataset (Penedo et al. 2024) (for LM and common-sense reasoning tasks) or C4 dataset (Raffel et al. 2020) (for scaling patterns). We use model",
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+ "type": "page_number",
+ "text": "15",
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+ "img_path": "images/23e8a3c068128a8a12d8568ecdde19b54a9ecb901162d0c610604a0abdd79fbe.jpg",
+ "image_caption": [
+ "Figure 3: Scaling patterns when increasing (Left) model size, (Middle) sequence length (model size = 340M) (3) (Right) sequence length (model size = 760M) on C4 dataset."
+ ],
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+ "img_path": "images/bb38c6f7c2bb840b3f533a645aa391e2005cae92f583aa262a8d6028d4e35f08.jpg",
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+ {
+ "type": "image",
+ "img_path": "images/1e71d292b87b5b94660e97445a73eff75c18a71916cb028bc365fee679360b73.jpg",
+ "image_caption": [],
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+ "type": "text",
+ "text": "sizes of 120M, 340M, 760M, and 1.3B parameters. We train small models (120M and 340M) on 15B tokens sampled from the dataset, the medium size model (760M) on 30B tokens, and the large model on 100B tokens. Baseline results are reported by Behrouz et al. (2024c).",
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+ "type": "text",
+ "text": "6.1 Language Modeling and Common-sense Reasoning",
+ "text_level": 1,
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+ "text": "We follow recent studies (Behrouz et al. 2024c; Yang et al. 2024a,c) and first focus on the perplexity in language modeling and also commonsense reasoning tasks. The results for MEMORA, YAAD, MONETA and also baselines with size of 340M, 760, and 1.3B are reported in Table 2. All of our variants outperforms all the baselines including Transformer++, modern linear recurrent models and hybrid methods. The superior performance compared to hybrid models is particularly important as all of our variants are pure recurrent (attention-free). Among the three variants of MirAS, while MONETA achieves slightly weaker performance than MEMORA, and YAAD, the other two variants are close and depending on the task and model size, the best model can vary.",
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+ "type": "text",
+ "text": "6.2 Scaling Pattern",
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+ {
+ "type": "text",
+ "text": "To evaluate the scaling pattern of models and for comparing them with baseline, in this section, we plot their performance with varying the model size and the context window.",
+ "bbox": [
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+ "page_idx": 15
+ },
+ {
+ "type": "text",
+ "text": "Context Length. We first vary the training context length from 2K to 32K for two versions of our model with size 340M and 760M. The results are reported in Figure 3 (Middle and Right). All three variants of Miras scales better than state-of-the-art baselines when increasing the context length. We attribute this superior performance to: (1) expressive memory architecture. Contrary to baselines like Mamba2 and GSA that uses vector- and matrix-valued memory, our variants are using 2-layer MLPs with more expressive power to learn from longer sequences. (2) The choice of retention gate and attentional bias: All of our three variants go beyond the standard attentional biases and retention gates. These choices can help the memory to better manage its fixed-size capacity.",
+ "bbox": [
+ 109,
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+ "page_idx": 15
+ },
+ {
+ "type": "text",
+ "text": "Model Size. We also report the #FLOPs vs. perplexity of our models and baselines in Figure 3 (Left). All three variants outperforms all baselines given almost the same budget of FLOPs. These results, once again support the importance of powerful memory design.",
+ "bbox": [
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+ "page_idx": 15
+ },
+ {
+ "type": "text",
+ "text": "6.3 Needle In Haystack",
+ "text_level": 1,
+ "bbox": [
+ 112,
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+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "text",
+ "text": "To evaluate the effective context window of our models and baselines, we use needle-in-haystack task. In this task, we evaluate the model on retrieving a piece of information (i.e., the \"needle\") from long distractor texts (i.e., the \"haystack\"). We focus on the Single NIAH (S-NIAH) task from RULER benchmark (Hsieh et al. 2024) and evaluate our models and baselines on sequences with length 1K, 2K, 4K, and 8K. The results are reported in Table 3. All our variants outperforms all the baselines with a considerable margin. Interestingly, MONETA shows better performance than others when the data is synthetic noise (S-NIAH-PK). This observation validates the effectiveness of $p$ -norm objective and retention gates as they are more robust to noise.",
+ "bbox": [
+ 109,
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+ "page_idx": 15
+ },
+ {
+ "type": "page_number",
+ "text": "16",
+ "bbox": [
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+ },
+ {
+ "type": "table",
+ "img_path": "images/7f3725d584d1950412336fd521c5fe3fd51caa25f4aa98854ba39b2420f74a32.jpg",
+ "table_caption": [
+ "Table 2: Performance of MIRAS's variants and baselines on language modeling and common-sense reasoning tasks. Hybrid models are marked with *. The best results of simple and hybrid models are highlighted. In largest scale, we compare our simple models with even hybrid models and highlight the best results."
+ ],
+ "table_footnote": [],
+ "table_body": "| Model | Memory Architecture | Attentional Bias | Retention Gate† | Memory Algorithm | Memory Write Operation |
| Shallow Memory |
| RetNet (2023) | Vector | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| Transformer (2017) | Matrix | L2 | - | Nonparametric | Mt=Mt-1∪{kt, vt} |
| LA (2021) | Matrix | Dot-Product | - | GD | Mt=Mt-1+vtktT |
| DFW | Matrix | Dot-Product | L2 | GD | Mt=(βtαT) ⊙ Mt-1+vtktT |
| Lightening Attention (2025) | Matrix | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| GLA (2024) | Matrix | Dot-Product | L2 | GD | Mt=Diag(αt)Mt-1+vtktT |
| Mamba (2024) | Matrix | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| HGRN2 (2024) | Matrix | L1 | L2 | GD | Mt=Diag(αt)Mt-1+vt(1-αt)T |
| DeltaNet (2017) | Matrix | L2 | - | GD | Mt=(I-βtktkT)Mt-1+βtvtktT |
| Longhorn (2024) | Matrix | L2 | - | Implicit GD | Mt=(I-βtktkT)Mt-1+(βt1+ktkβt)xtkT |
| TTT-Linear (2024) | Matrix | L2 | - | GD | Mt=Mt-1-η∇L(Mt-1, xt) |
| Gated DeltaNet (2024) | Matrix | L2 | L2 | GD | Mt=(αt(I-βtktkT))Mt-1+βtvtktT |
| RWKV-7 (2025) | Matrix | L2 | L2 | GD | Mt=diag(αt)(I-βtktkT)Mt-1+βtvtktT |
| DeltaProduct (2025) | Matrix | L2 | L2 | MGD* | Mt=(αtΠi=1n(I-βt,ikt,i)T)Mt-1+Σj=1nΠi=j(I-βt,ivtj,kj,i) |
| Deep Memory |
| TTT-MLP (2024) | 2-layer MLP | L2 | - | GD | Mt=Mt-1-η∇L(Mt-1;kt, vt) |
| Titans-LMM (2024) | k-layer MLP | L2 | L2 | GD + Momentum | Mt=αMt-1-St, where St=ηSt-1-θt∇L(Mt-1;kt, vt) |
| MONETA (ours) | 2-layer MLP | Lp | Lq | GD | At=AtA1-ηt∇lp(Wt-1;kt, vt), Wt=At/||At||q-2 |
| YAAD (ours) | 2-layer MLP | Huber | L2 | GD | Wt=atWt-1-(ηt∇ε2(Wt-1;kt, vt) if ||M(kt)-vt|≤δt, ηtδt∇ε1(Wt-1;kt, vt) Otherwise. |
| MEMORA (ours) | 2-layer MLP | L2 | KL | GD | Wt=Softmax(αt log(Wt-1)-ηt∇ε2(Wt-1;kt, vt)) |
"
+ },
+ {
+ "type": "table_footnote",
+ "bbox": [
+ 0.123,
+ 0.503,
+ 0.335,
+ 0.513
+ ],
+ "angle": 0,
+ "content": "* is using multiple rounds of GD per token."
+ },
+ {
+ "type": "table_footnote",
+ "bbox": [
+ 0.123,
+ 0.513,
+ 0.891,
+ 0.525
+ ],
+ "angle": 0,
+ "content": "For the sake of clarity, we use L2 for all modified L2-like regularizations. However, in fact, only Titans and RWKV-7 are using L2 retention gate (see Section 4)"
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.123,
+ 0.503,
+ 0.891,
+ 0.525
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.552,
+ 0.92,
+ 0.72
+ ],
+ "angle": 0,
+ "content": "Deep Memory Module: Titans and Test Time Training. To overcome the limited memory and to extend the effective context length of deep sequence models, more recent studies focus on a new generation of architectures with deep memory module (Behrouz et al. 2024c; Sun et al. 2024). These architectures are built on the meta-learning perspective, where the memory is an MLP architecture that is updated using gradient descent (with momentum) (Behrouz et al. 2024c; Sun et al. 2024). Sun et al. (2024) further provide a unifying perspective that how linear and softmax attention are respectively parametric and non-parametric solutions of (kernel) regression loss but consider other modern linear RNNs outside of this class of models. Recently, in a concurrent work to ours, Wang et al. (2025) show that with additional simplification of modern RNNs (e.g., RetNet (Sun et al. 2023), Mamba (Dao et al. 2024)) they approximately place in the same class of models that internally optimize regression loss. It, however, still remains unanswered that \"What is the underlying design framework behind these sequence models that can accurately unify existing architectures?\" Moreover, the role of forget gates and its alternative choices in modern sequence models is surprisingly less explored."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.111,
+ 0.741,
+ 0.719,
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+ ],
+ "angle": 0,
+ "content": "3 Associative Memory, Attentional Bias, and Retention"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.772,
+ 0.92,
+ 0.865
+ ],
+ "angle": 0,
+ "content": "Associative memory, which is an inseparable component of learning in humans (Terry 2017), has been the inspiration for many artificial neural architectures in the literature (Behrouz et al. 2024c; Hopfield 1982; Neil et al. 2017). These studies, however, define instances of the concept of associative memory, limiting the architecture to a specific class of similarity metrics between entities (i.e., keys and values). That is, broadly speaking, associative memory is an operator that maps a set of keys \\( K \\) to a set of values \\( V \\), and so to learn the underlying mapping patterns in data, it requires an objective that targets a type of memory and measures the quality of learned mappings:"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.87,
+ 0.92,
+ 0.903
+ ],
+ "angle": 0,
+ "content": "Definition 3.1 (Associative Memory and Attentional Bias). Given a set of keys \\(\\mathcal{K} \\subseteq \\mathbb{R}^{d_k}\\) and values \\(\\mathcal{V} \\subseteq \\mathbb{R}^{d_o}\\), associative memory is an operator \\(\\mathcal{M}: \\mathcal{K} \\to \\mathcal{V}\\). Learning the mapping of associative memory is based on an objective \\(\\mathcal{L}\\), called"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.512,
+ 0.938,
+ 0.523,
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+ ],
+ "angle": 0,
+ "content": "4"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.093,
+ 0.759,
+ 0.109
+ ],
+ "angle": 0,
+ "content": "Attentional Bias, that determines the type of memory and its tendency to prioritize some events:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.407,
+ 0.12,
+ 0.92,
+ 0.142
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {M} ^ {*} = \\arg \\min _ {\\mathcal {M}} \\quad \\mathcal {L} (\\mathcal {M} (\\mathcal {K}); \\mathcal {V}). \\tag {4}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.161,
+ 0.304,
+ 0.175
+ ],
+ "angle": 0,
+ "content": "A few remarks are in order:"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.184,
+ 0.921,
+ 0.23
+ ],
+ "angle": 0,
+ "content": "Remark 1. When we parameterize the memory with parameter \\( W \\), we use \\( \\mathcal{M}(W, \\mathbf{k}) \\). In this parametric setting, the optimization problem in (4) should be performed over the parameter \\( W \\). Furthermore, in the parametric setup, we might use an additional regularization \\( \\mathcal{R}(W) \\) to control the retaining of the past data."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.237,
+ 0.922,
+ 0.297
+ ],
+ "angle": 0,
+ "content": "Remark 2. Learning the mapping between keys and values (Equation 4) is a meta-learning problem, in which the attentional bias is optimized in the inner-loop and all other parameters of the neural network (e.g., linear projections, convolutions, etc.) are optimized in the outer-loop. Therefore, the model learns how to store the data into its parameters at test time (Behrouz et al. 2024c; Sun et al. 2024)."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.112,
+ 0.316,
+ 0.792,
+ 0.334
+ ],
+ "angle": 0,
+ "content": "3.1 Learning to Memorize and to Retain Through the Lens of Optimization"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.34,
+ 0.92,
+ 0.386
+ ],
+ "angle": 0,
+ "content": "Definition 3.1 translates the design of a neural architecture based on the concept of associative memory to learning the underlying mapping between keys and values, by minimizing an objective \\(\\mathcal{L}\\). To optimize Equation 4, one simple approach is to utilize the idea of gradient descent. Specifically, given a new pair of keys and values, we update the memory as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.409,
+ 0.398,
+ 0.92,
+ 0.415
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = W _ {t - 1} - \\eta_ {t} \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right), \\tag {5}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.426,
+ 0.92,
+ 0.502
+ ],
+ "angle": 0,
+ "content": "where, for simplicity, we use the definition \\(\\ell(W_{t-1}; \\mathbf{k}_t, \\mathbf{v}_t) \\coloneqq \\mathcal{L}(\\mathcal{M}(W; \\mathbf{k}_t), \\mathbf{v}_t)\\). Behrouz et al. (2024c) re-interpret the formulation as a momentary surprise metric, where the model memorizes tokens that violates the expectation of the objective (i.e., being surprising to the memory). Although the choice of objective is an important step to fully interpret Equation 5 (which we discuss in detail in Section 5), there are different viewpoints to interpret this update rule in its general format, which later can help us to go beyond existing architectures:"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.112,
+ 0.52,
+ 0.767,
+ 0.538
+ ],
+ "angle": 0,
+ "content": "3.2 Viewpoint 1: Online Regression and Follow-The-Regularized-Leader"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.545,
+ 0.816,
+ 0.56
+ ],
+ "angle": 0,
+ "content": "Equation (5) can be viewed as one step of online gradient descent over the sequence of the loss functions"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.362,
+ 0.572,
+ 0.92,
+ 0.589
+ ],
+ "angle": 0,
+ "content": "\\[\n\\ell \\left(W; \\mathbf {k} _ {1}, \\mathbf {v} _ {1}\\right), \\ell \\left(W; \\mathbf {k} _ {2}, \\mathbf {v} _ {2}\\right), \\dots , \\ell \\left(W; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right), \\dots . \\tag {6}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.6,
+ 0.92,
+ 0.646
+ ],
+ "angle": 0,
+ "content": "It is well known that the online gradient descent can be viewed as a special case of Follow-The-Regularized-Leader (FTRL) algorithm with a special choice of loss functions (Shalev-Shwartz et al. 2012, Chapter 2) and (Hazan et al. 2016). Specifically, assuming \\( W_0 = 0 \\), the update rule in (5) is equivalent to"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.318,
+ 0.658,
+ 0.92,
+ 0.697
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = \\arg \\min _ {W} \\quad \\sum_ {i = 1} ^ {t} \\left\\langle W - W _ {i - 1}, \\nabla \\ell \\left(W _ {i - 1}; \\mathbf {k} _ {i}, \\mathbf {v} _ {i}\\right) \\right\\rangle + \\frac {1}{2 \\eta} \\| W \\| _ {2} ^ {2}, \\tag {7}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.707,
+ 0.919,
+ 0.755
+ ],
+ "angle": 0,
+ "content": "where the term \\(\\langle W - W_{i-1}, \\nabla \\ell(W_{i-1}; \\mathbf{k}_i, \\mathbf{v}_i) \\rangle\\) is the local linear approximation of the original loss at time \\(i\\) and the second term is a regularization term. While the first part \\(\\sum_{i=1}^{t} \\langle W - W_{i-1}, \\nabla \\ell(W_{i-1}; \\mathbf{k}_i, \\mathbf{v}_i) \\rangle\\) measures how well can the memory learn all the past tokens, the second term \\(\\frac{1}{2\\eta} \\|W\\|_2^2\\) penalizes the memory update with respect to the size of memory."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.761,
+ 0.922,
+ 0.837
+ ],
+ "angle": 0,
+ "content": "Equation (7) uses linear approximation of the loss function and quadratic regularization. We can, however, in principle use other approximations of the loss function as well as other regularization functions, as used in the past in online optimization (Hazan et al. 2016; Shalev-Shwartz et al. 2012) or in general optimization (Miral 2015; Razaviyayn et al. 2013). Such changes are the idea behind the development of other optimization algorithms such mirror descent. More specifically, we can generalize the update rule in (7) to the form:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.351,
+ 0.855,
+ 0.92,
+ 0.917
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = \\arg \\min _ {W \\in \\mathcal {W}} \\underbrace {\\sum_ {i = 1} ^ {t} \\widehat {\\ell_ {i}} (W ; \\mathbf {k} _ {i} , \\mathbf {v} _ {i})} _ {\\text {A t t e n t i o n a l B i a s}} + \\underbrace {\\frac {1}{\\eta_ {t}} \\mathcal {R} _ {t} (W)} _ {\\text {M e m o r y S t a b i l i t y}}. \\tag {FTRLViewpoint}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.512,
+ 0.938,
+ 0.522,
+ 0.948
+ ],
+ "angle": 0,
+ "content": "5"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.092,
+ 0.921,
+ 0.172
+ ],
+ "angle": 0,
+ "content": "In this update rule, the term \\(\\sum_{i=1}^{t} \\widehat{\\ell}_i(W; \\mathbf{k}_i, \\mathbf{v}_i)\\) aims at memorizing the tokens at test time, while the term \\(\\mathcal{R}_t(W)\\) regularizes the learning dynamics and take the size of the memory into account when updating it by a new incoming data. Choosing different loss functions \\(\\widehat{\\ell}_i(W; x_i)\\) and the regularization term \\(\\frac{1}{\\eta_t} \\mathcal{R}_t(W)\\) can lead to different algorithms such as (online) gradient descent or mirror descent. In this generalization, \\(\\eta_t\\) to can be data-dependent. Moreover, we will allow imposing constraint \\(\\mathcal{W}\\) on the choice \\(W\\)."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.112,
+ 0.188,
+ 0.861,
+ 0.206
+ ],
+ "angle": 0,
+ "content": "3.3 Viewpoint 2: Learning the Latest Token While Retaining Previous Information"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.213,
+ 0.921,
+ 0.259
+ ],
+ "angle": 0,
+ "content": "Another way to interpret the update rule (5) is to view it as learning from the latest key-value pair \\((\\mathbf{k}_i, \\mathbf{v}_i)\\) (via using its gradient or surprise metric), while staying close to the previous state \\(W_{t-1}\\) to retain the previously memorized tokens. Formally, (5) is equivalent to"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.301,
+ 0.27,
+ 0.731,
+ 0.3
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = \\arg \\min _ {W} \\left\\langle W - W _ {t - 1}, \\nabla \\ell (W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) \\right\\rangle + \\frac {1}{2 \\eta_ {t}} \\left\\| W - W _ {t - 1} \\right\\| _ {2} ^ {2}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.311,
+ 0.919,
+ 0.342
+ ],
+ "angle": 0,
+ "content": "The first term locally approximates \\(\\ell(W; \\mathbf{k}_t, \\mathbf{v}_t)\\) around the previous state \\(W_{t-1}\\), while the last term regularizes deviations from \\(W_{t-1}\\). This form can generalize to"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.255,
+ 0.374,
+ 0.921,
+ 0.415
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = \\arg \\min _ {W \\in \\mathcal {W}} \\underbrace {\\widetilde {\\ell_ {t}} (W ; \\mathbf {k} _ {t} , \\mathbf {v} _ {t})} _ {\\text {A t t e n t i o n a l B i a s}} + \\underbrace {\\operatorname {R e t} _ {t} (W , W _ {t - 1})} _ {\\text {R e t e n t i o n}}, \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\text {(L e a r n i n g - R e t a i n i n g V i e w p o i n t)}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.425,
+ 0.92,
+ 0.472
+ ],
+ "angle": 0,
+ "content": "where the term \\(\\widetilde{\\ell_t} (W;\\mathbf{k}_t,\\mathbf{v}_t)\\) is an approximation of \\(\\ell (W;\\mathbf{k}_t,\\mathbf{v}_t)\\) and minimizing it corresponds to Learning from the new concepts \\((\\mathbf{k}_t,\\mathbf{v}_t)\\). The second term \\(\\mathrm{Ret}_t(W,W_{t - 1})\\) regularizes the changes in \\(W\\) to make the learning dynamics stable and to retain previously learned knowledge. This Retention function may have local and global components:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.351,
+ 0.482,
+ 0.68,
+ 0.537
+ ],
+ "angle": 0,
+ "content": "\\[\n\\operatorname {R e t} _ {t} \\left(W, W _ {t - 1}\\right) = \\underbrace {\\frac {1}{\\eta_ {t}} \\mathrm {D} _ {t} \\left(W , W _ {t - 1}\\right)} _ {\\text {L o c a l R e t e n t i o n}} + \\underbrace {\\frac {1}{\\alpha_ {t}} \\mathrm {G} _ {t} \\left(W\\right)} _ {\\text {G l o b a l R e t e n t i o n}}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.546,
+ 0.92,
+ 0.637
+ ],
+ "angle": 0,
+ "content": "Here, the term \\( \\mathrm{D}_t(W, W_{t-1}) \\), which is a premetric that controls the deviations from \\( W_{t-1} \\), aims at retaining previously learned knowledge. The coefficient \\( \\eta_t \\) can be viewed as a meta in-context learning rate, where larger values of \\( \\eta_t \\) leads to learning more from new concepts, while allowing higher forgetting of previously learned concepts. The second term is a global retention that controls the change of the memory with respect to its size. The special instances of the above viewpoint (e.g., without global retention, with implicit closed-form solution, and/or with limited memory structure) have been the motivation behind some of the recent studies such as Liu et al. (2024a)."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.113,
+ 0.655,
+ 0.549,
+ 0.673
+ ],
+ "angle": 0,
+ "content": "3.4 Further Discussions on the Two Viewpoints"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.68,
+ 0.92,
+ 0.755
+ ],
+ "angle": 0,
+ "content": "The (FTRL Viewpoint) and (Learning-Retaining Viewpoint) are connected through the lens of online optimization. For example, as discussed above, by choosing linear approximation of the loss and quadratic regularization/retention, they can both cover online gradient descent update in (5) as a special case. One straightforward way to make the connection explicit is by defining the premetric \\(\\mathrm{D}_t(W;W^{\\prime})\\) based on the previous loss functions and the regularization, as described in Proposition 3.2 below:"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.762,
+ 0.92,
+ 0.826
+ ],
+ "angle": 0,
+ "content": "Proposition 3.2. Let \\(\\eta_t = \\eta\\) and define \\(h_t(W) \\coloneqq \\sum_{i=1}^{t-1} \\widehat{\\ell}_i(W; \\mathbf{k}_i, \\mathbf{v}_i) + \\frac{1}{\\eta} R(W)\\). Assume \\(\\mathcal{W} = \\mathbb{R}^d\\) and the function \\(h_t(W)\\) is strictly convex in \\(W\\) and let \\(\\mathcal{D}_h(\\cdot, \\cdot)\\) be the Bregman divergence defined by function \\(h(\\cdot)\\), i.e., \\(\\mathcal{D}_h(W, W') = h(W) - h(W') - \\langle \\nabla h(W'), W - W' \\rangle\\). Set \\(Ret_t(W, W') = \\mathcal{D}_h(W, W')\\) and \\(\\widetilde{\\ell}_t(W; x_t) = \\widehat{\\ell}_t(W; x_t)\\) in (Learning-Retaining Viewpoint). Then, the update rule in (Learning-Retaining Viewpoint) is equivalent to the update rule in (FTRL Viewpoint)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.832,
+ 0.92,
+ 0.892
+ ],
+ "angle": 0,
+ "content": "We provide the proof in Appendix B. The above proposition shows that (Learning-Retaining Viewpoint) can also explain the approaches obtained by (FTRL Viewpoint), under some mild assumptions. Hence, (Learning-Retaining Viewpoint) may be seen as a more general version. This is why we focus on this viewpoint in most of our derivations in the next sections."
+ },
+ {
+ "type": "page_number",
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+ "angle": 0,
+ "content": "6"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Remark 3. Given the above viewpoint, we can see that even by using additional global regularization there is no memory erasing or forgetting process (a common term in modern architectures (Behrouz et al. 2024c; Yang et al. 2024a)) but the model might decide to not retain the past state of the memory. Interestingly, this observation also matches the human memory process, where brain does not erase memories but they might become inaccessible due to retrieval failures (Robertson 2002). Therefore, instead of calling it a forget gate, later on, we use \"Retention Gate\" to refer to this term."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.176,
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+ ],
+ "angle": 0,
+ "content": "Remark 4. As we discuss in Section 4 and summarize in Table 1, most existing modern sequence models are optimizing associative memory objective (attentional bias in Equation 4) using gradient descent. Therefore, to provide further intuition about the connection of existing sequence models as well as their online learning interpretations, we discuss the above two viewpoints that are limited to gradient descent-based update rules. Our initial definition of attentional bias and associative memory in Equation 4, however, is broader and can be optimized by any optimization algorithm (e.g., even Newton's method, or non-parametric solutions)."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "4 MirAs: Learning to Memorize with Robust and Expressive Memory"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.32,
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+ "angle": 0,
+ "content": "Building upon our definition of associative memory, attentional bias, and previous viewpoints, we present MIRAs framework that not only accurately unifies existing backbone architectures but it also provides insights on how to design the next generation of sequence models. As discussed earlier in Section 3, learning an associative memory can be interpreted as a meta-learning task, in which the associative memory learns how to compress and store data into its parameters at test time. The architecture of the memory in such tasks is particularly important as in longer contexts, the expressivity of the memory structure can limit its ability to learn the underlying patterns. Therefore, the first choice to design a sequence model is the structure of the memory. Given the structure of the memory, parameterized by a set of parameters \\( W \\), as discussed earlier, we aim to minimize a loss function \\( \\ell(W; \\cdot, \\cdot) \\) with a retention regularizer \\( \\mathrm{Ret}(\\cdot) \\) via a learning algorithm (e.g., gradient descent). Accordingly, MIRAs requires four design choices:"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.118,
+ 0.463,
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+ "angle": 0,
+ "content": "1. Memory Structure: This choice specifies the architecture of the memory. For example, this architecture can be a vector, a linear function, a Multilayer Perceptron (MLP) layer, or even more complex structures. We may restrict the choice of \\( W \\) to be within a certain region, e.g., \\( W \\) to lie within an \\( L_{2} \\) ball to avoid infinite values or unstable training."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.118,
+ 0.516,
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+ ],
+ "angle": 0,
+ "content": "2. Attentional Bias: A key choice is the attentional bias objective \\(\\mathcal{L}(\\cdot)\\) in Equation 4. We can even consider different approximations of the loss function, (e.g., \\(\\widehat{\\ell} (\\cdot ,\\cdot)\\) in (FTRL Viewpoint) or \\(\\widetilde{\\ell} (\\cdot ,\\cdot)\\) in (Learning-Retaining Viewpoint)). The choice of attentional bias determines how memory memorizes the context, maps the inputs, and prioritizes the events."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "3. Memory Stability and Retention: Another key choice is the retention regularizer \\(\\mathcal{R}(\\cdot)\\) (e.g., \\(\\mathcal{R}_t(\\cdot)\\) in (FTRL Viewpoint) and \\(\\mathrm{Ret}_t(\\cdot)\\) in (Learning-Retaining Viewpoint)). In parametric setups, this choice balances learning with retention of past state. An effective retention gate is key to the good performance in long context tasks."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "4. Memory Algorithm: Finally, this choice specifies the learning algorithm that we use to optimize the memory objective. One may use gradient descent, gradient descent with momentum, or any other algorithm (including finding non-parametric solutions)."
+ },
+ {
+ "type": "list",
+ "bbox": [
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+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
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+ "angle": 0,
+ "content": "The above choices are major design choices for designing backbone sequence models in neural architectures. There are, however, minor decisions that can distinguish models; i.e., data-dependent or independent parameters, scalar or channel-wise learning rate/retaining gate, etc. Next, we discuss the overview of how existing architectures fit into MIRAS framework."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.748,
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+ "angle": 0,
+ "content": "RNNs with Hebbian Rule. The first generation of modern recurrent architectures (e.g., Linear attention (Katharopoulos et al. 2020), RetNet (Sun et al. 2023), Mamba (Gu et al. 2024), and GLA (Yang et al. 2024b)) are based on Hebbian-like (e.g., gated Hebbian) learning rule (Hebb 2005). We let attentional bias be the dot product similarity. That is, given a memory \\(\\mathcal{M} \\in \\mathbb{R}^{d \\times n}\\) and \\(\\mathbf{k}, \\mathbf{v} \\in \\mathbb{R}^d\\), we define \\(\\tilde{\\ell}_t \\coloneqq -2\\langle \\mathcal{M}_t \\mathbf{k}_t, \\mathbf{v}_t \\rangle\\) and local retention as \\(\\mathrm{Ret}_t(\\mathcal{M}, \\mathcal{M}_{t-1}) = \\| \\mathcal{M}_t - \\alpha \\mathcal{M}_{t-1} \\|_F^2\\). Using Equation Learning-Retaining Viewpoint and gradient descent as the optimizer (i.e., memory learning algorithm), the memory update rule is:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.442,
+ 0.845,
+ 0.92,
+ 0.863
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {M} _ {t} = \\alpha \\mathcal {M} _ {t - 1} + \\mathbf {v} _ {t} \\mathbf {k} _ {t} ^ {\\top}. \\tag {8}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.868,
+ 0.92,
+ 0.915
+ ],
+ "angle": 0,
+ "content": "When (1) \\(\\alpha = 1\\), memory update is equivalent to Linear Attention (LA) (Katharopoulos et al. 2020); (2) \\(\\alpha \\in \\mathbb{R}\\) is a learnable parameter, resulting architecture is either lightening attention (\\(n > 1\\)) (Li et al. 2025) or RetNet (\\(n = 1\\)) (Sun et al. 2023); and (3) \\(\\alpha_{t} \\in \\mathbb{R}\\) are data-dependent learnable parameters, resulting sequence model is Mamba2 (Dao et al. 2024)."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.512,
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+ "angle": 0,
+ "content": "7"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.093,
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+ ],
+ "angle": 0,
+ "content": "RNNs with Delta Rule. To improve the memory management and to enhance the memory capacity of the above group, several studies suggest using delta rule (Neil et al. 2017; Schlag et al. 2021) as the learning algorithm in recurrent neural networks (e.g., DeltaNet (Schlag et al. 2021), Longhorn (Liu et al. 2024a), and RWKV7 (Peng et al. 2025b)). In this part, we recall that where \\(\\mathcal{M} \\in \\mathbb{R}^{d \\times n}\\), delta rule is equivalent to optimizing MSE objective \\(\\| \\mathcal{M}_t \\mathbf{k}_t - \\mathbf{v}_t \\|_2^2\\) with \\(\\mathrm{Ret}_t(\\mathcal{M}, \\mathcal{M}_{t-1}) = \\| \\mathcal{M}_t - \\alpha \\mathcal{M}_{t-1} \\|_F^2\\) as local retention, and stochastic gradient descent as optimizer: (\\(\\eta_t\\) is defined in Equation Learning-Retaining Viewpoint)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.398,
+ 0.195,
+ 0.92,
+ 0.213
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {M} _ {t} = \\alpha \\left(\\mathbf {I} - \\eta_ {t} \\mathbf {k} _ {t} \\mathbf {k} _ {t} ^ {\\top}\\right) \\mathcal {M} _ {t - 1} + \\mathbf {v} _ {t} \\mathbf {k} _ {t} ^ {\\top}. \\tag {9}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.224,
+ 0.921,
+ 0.268
+ ],
+ "angle": 0,
+ "content": "When (1) \\(\\alpha = 1\\), memory update is equivalent to DeltaNet (Schlag et al. 2021); and (2) \\(\\alpha_{t} \\in \\mathbb{R}^{m}\\) are data-dependent learnable parameters, resulting sequence model is either Gated DeltaNet (Yang et al. 2024a) (\\(m = 1\\)), or RWKV7 (Peng et al. 2025b) (\\(m = d\\)). Therefore, RNNs with delta rule are special instances of MIRAS."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.282,
+ 0.922,
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+ ],
+ "angle": 0,
+ "content": "Beyond Delta Rule. As discussed earlier, while delta rule with its value replacement strategy is more powerful than Hebbian-like learning rules, it suffers from theoretical limitations (Irie et al. 2023) and achieves moderate performance in practice (Yang et al. 2024c). Therefore, several studies have focused on update rules beyond delta rule. Recently, Titans (Behrouz et al. 2024c) suggests using non-linear MSE objective of \\(\\| \\mathcal{M}_t(\\mathbf{k}_t) - \\mathbf{v}_t\\| _2^2\\) with both local and global retention of \\(\\mathrm{D}_t = \\| W_t - W_{t - 1}\\| _F^2\\) and \\(\\mathrm{G}_t = \\| W_t\\| _2^2\\) and optimize it with gradient descent with momentum \\(^2\\). Therefore, Titans-LMM is a special instance of MIRAs, where we use the abovementioned attentional bias and retention regularizations, and gradient descent with momentum as the optimizer."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.396,
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+ ],
+ "angle": 0,
+ "content": "Another example of such models is Mesa-layer, in which the model uses \\(\\sum_{i=1}^{t} \\|\\mathcal{M}_{t}(\\mathbf{k}_{i}) - \\mathbf{v}_{i}\\|_{2}^{2}\\) as the attentional bias objective with \\(\\|\\mathcal{M}_{t}\\|_{2}^{2}\\) as the retention regularization. Since these models use Newton's method to optimize such an objective, they provide a more expressive update rule than delta rule. We further discuss a set of new learning algorithms beyond delta rule in Section 5."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.47,
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+ ],
+ "angle": 0,
+ "content": "Attention. As discussed by Sun et al. (2024), softmax attention is a non-parametric solution of \\(\\ell_2\\)-MSE loss function (i.e., \\(\\| W\\mathbf{k} - \\mathbf{v}\\| _2^2\\)) with Nadaraya-Watson estimator. Therefore, softmax attention is an instance of MIRAS, when we find the non-parametric solution to the MSE loss with Nadaraya-Watson estimator, without retention."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "5 Beyond Existing Attentional Biases and Retention Gates"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
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+ "angle": 0,
+ "content": "As discussed in the previous section, existing work focuses only on linear/quadratic choices for the attentional bias or retention gate. In particular, the loss function \\( L(\\mathcal{M}(\\mathbf{k}_t),\\mathbf{v}_t) \\) is defined as \\( L(\\mathcal{M}(\\mathbf{k}_t),\\mathbf{v}_t) = c_t\\| \\mathcal{M}(\\mathbf{k}_t) - \\mathbf{v}_t\\|^2 \\) for some (learnable) constant \\( c_{t} \\) in prior work. Also the regularization term \\( R_{t}(W) \\) or the parametric \\( D_{t} \\) is considered as a quadratic/linear function. In addition, almost all prior work considers \\( W \\) to be the entire \\( \\mathbb{R}^d \\) space. However, in general there could be various choices for all the three aforementioned design choices. To illustrate the potential and flexibility of our designed framework, here, we review some of the potential design choices for attentional bias and retention gate in MirAS. For the sake of clarity, we discuss all these attentional bias and memory retention gates based on using gradient descent as the optimizer, and so based on the provided two view points. However, these attentional bias objectives and retention regularizers can be directly used in Equation 4 and optimized by using any other optimization algorithms, resulting in different update rules."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "5.1 Alternative Attentional Biases"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Variant 1: \\(\\ell_p\\)-Attentional Bias. As discussed in the main body, attentional bias defines the \"similarity metric\" and measures how well memory can recall the value, given its corresponding key. Although \\(\\ell_2\\) regression loss often is a natural choice, it is sensitive to noise in the data. A natural extension is to use \\(\\ell_p\\)-norm class of objectives. That is, let \\(\\mathcal{M}\\) be the memory, \\(\\mathbf{k}\\) be the keys, and \\(\\mathbf{v}\\) be the values, we define \\(\\ell_p\\)-attentional bias as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.394,
+ 0.841,
+ 0.92,
+ 0.861
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {L} \\left(\\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right); \\mathbf {v} _ {t}\\right) = \\| \\mathcal {M} \\left(\\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {p} ^ {p}, \\tag {10}\n\\]"
+ },
+ {
+ "type": "page_footnote",
+ "bbox": [
+ 0.112,
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+ ],
+ "angle": 0,
+ "content": "The retention gate (forget gate) in Titans is different from Mamba2 and Gated DeltaNet that we discussed above. The main difference comes from the case of full memory erase. While Mamba2 gating removes the entire memory and treats the next token as the first ever seen data, Titans use a \"cold start\" strategy and use the previous state of the memory to measure the surprise of the incoming token before fully erasing the memory."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "8"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.093,
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+ 0.139
+ ],
+ "angle": 0,
+ "content": "where \\( p \\in \\mathbb{R}^{\\geq 1} \\) and \\( \\| . \\|_p \\) is the \\( p \\)-norm. Although depending on the distribution of the data, we might want to use different values of \\( p \\) (see Section 6), different values of \\( p \\) can result in memory architectures with interesting properties. For the sake of simplicity, let memory be a matrix, i.e., \\( W \\in \\mathbb{R}^{m \\times d} \\) and \\( \\mathcal{M}(W, \\mathbf{k}_t) = W\\mathbf{k}_t \\), the closed form can be derived as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.243,
+ 0.149,
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+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = W _ {t} - \\eta_ {t} \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) = W _ {t} - p \\eta_ {t} \\left(\\operatorname {S i g n} \\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\odot | W \\mathbf {k} _ {t} - \\mathbf {v} _ {t} | ^ {p - 1}\\right) \\mathbf {k} _ {t} ^ {\\top}. \\tag {11}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
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+ ],
+ "angle": 0,
+ "content": "Let \\(p = 1\\), the recurrence is simplified as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = W _ {t} - \\eta_ {t} \\operatorname {S i g n} \\left(W _ {t} \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {\\top}, \\tag {12}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "which means that the memory has only two values of \\(-1\\) and \\(1\\). We call this variation value-less associative memory, in which we store entities (keys) but map them into two extreme class of \\(-1\\) and \\(+1\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Remark 5. One of the critical challenges to use the above update rule is in the backpropagation process, in which \\(\\operatorname{Sign}(\\cdot)\\) and \\(|\\cdot|\\) are non-differentiable and so might cause unstable training. To overcome this issue, we use \\(\\operatorname{Sign}(x) \\approx \\tanh(\\alpha x)\\), and \\(|x| = \\sqrt{x^2 + \\epsilon}\\), as the smooth approximators of these functions."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "One simple interpretation for such behavior (i.e., value-less memory) is similar to the coping mechanism in humans (Loftus 1993), in which the memory does not store the values for extreme events. This interpretation of protective memory in extreme events motivates our next variant."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Variant 2: Huber Loss: Memory with Coping Mechanism. While \\(\\ell_2\\)-norm objective is a common choice for many statistical and machine learning tasks, it is known to be sensitive to outliers and extreme samples. This sensitivity extends to the use of \\(\\ell_2\\) loss for attentional bias. To address this and drawing motivation from robust regression literature, we suggest utilizing the Huber loss-type (Hastie et al. 2009; Huber 1992) as the attentional bias, thereby reducing the negative impact of the outlier data on the memory learning process."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
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+ "angle": 0,
+ "content": "We can apply Huber-type loss in three different ways: The first approach is to define the summation of the Huber loss across different coordinates as the total loss, i.e.,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.38,
+ 0.506,
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+ 0.54
+ ],
+ "angle": 0,
+ "content": "\\[\n\\ell (W; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) = \\sum_ {j} \\mathcal {H} (\\mathcal {M} (W, \\mathbf {k} _ {t}) _ {j} - \\mathbf {v} _ {t, j}),\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.551,
+ 0.92,
+ 0.579
+ ],
+ "angle": 0,
+ "content": "where \\(\\mathcal{M}(W,\\mathbf{k}_t)_j\\) and \\(\\mathbf{v}_{t,j}\\) denote the \\(j\\)-th coordinate of \\(\\mathcal{M}(W,\\mathbf{k}_t)\\) and \\(\\mathbf{v}_t\\) respectively. The function \\(\\mathcal{H}(\\cdot):\\mathbb{R}\\mapsto \\mathbb{R}\\) is the Huber loss defined as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.393,
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+ "angle": 0,
+ "content": "\\[\n\\mathcal {H} (a) = \\left\\{ \\begin{array}{l l} \\frac {1}{2} a ^ {2} & \\text {i f} | a | \\leq \\delta \\\\ \\delta \\left(| a | - \\frac {1}{2} \\delta\\right) & \\text {i f} | a | > \\delta . \\end{array} \\right. \\tag {13}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.615,
+ 0.92,
+ 0.647
+ ],
+ "angle": 0,
+ "content": "Utilizing this attentional bias can lead to various memory update rules. For example, for the matrix form memory \\(\\mathcal{M}(W,\\mathbf{k}_t) = W\\mathbf{k}_t\\), the update rule is given by"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.135,
+ 0.656,
+ 0.921,
+ 0.683
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = W _ {t - 1} - \\eta_ {t} \\left[ \\left(\\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {T}\\right) \\odot \\left(\\mathbf {I} \\left(\\left| W \\mathbf {k} _ {t} - \\mathbf {v} _ {t} \\right| \\leq \\delta_ {t}\\right) \\mathbf {1} ^ {\\top}\\right) + \\left(\\delta_ {t} \\operatorname {S i g n} \\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\mathbf {k} ^ {\\top}\\right) \\odot \\left(\\mathbf {I} \\left(\\left| W \\mathbf {k} _ {t} - \\mathbf {v} _ {t} \\right| > \\delta_ {t}\\right) \\mathbf {1} ^ {\\top}\\right) \\right] \\tag {14}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "In this formulation, the parameter \\(\\delta_t\\) decides the type of the memory used for each block of memory (\\(\\ell_2\\)-norm objective or value-less) based on the context, making the memory more robust to outliers."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
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+ ],
+ "angle": 0,
+ "content": "The second approach is to define the Huber-type loss based on the \\(\\ell_2\\) loss over all coordinates, i.e.,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.388,
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+ 0.774
+ ],
+ "angle": 0,
+ "content": "\\[\n\\ell (W; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) = \\mathcal {H} (\\| \\mathcal {M} (W, \\mathbf {k} _ {t}) - \\mathbf {v} _ {t} \\| _ {2}).\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.786,
+ 0.92,
+ 0.817
+ ],
+ "angle": 0,
+ "content": "For simplicity of derivations, assume matrix memory \\( M(W,\\mathbf{k}_t) = W\\mathbf{k}_t \\). Then using gradient descent for updating memory leads the memory update rule"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.27,
+ 0.827,
+ 0.921,
+ 0.868
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = W _ {t - 1} - \\eta_ {t} \\left\\{ \\begin{array}{l l} \\left(\\mathcal {M} \\left(W _ {t - 1}, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {T} & \\text {i f} \\| \\mathcal {M} \\left(W _ {t - 1}, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2} \\leq \\delta_ {t}, \\\\ \\delta_ {t} \\frac {\\left(\\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}\\right)}{\\| \\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2}} \\mathbf {k} _ {t} ^ {T} & \\text {O t h e r w i s e .} \\end{array} \\right. \\tag {15}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.878,
+ 0.92,
+ 0.909
+ ],
+ "angle": 0,
+ "content": "Again, in the form (15), the parameter \\(\\delta_t\\) decides the type of the memory used (\\(\\ell_2\\)-norm objective or normalized version) based on the context, making the memory more robust to outliers."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.512,
+ 0.938,
+ 0.522,
+ 0.948
+ ],
+ "angle": 0,
+ "content": "9"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.093,
+ 0.92,
+ 0.123
+ ],
+ "angle": 0,
+ "content": "Finally, in the third approach, we present a smooth mixture method, in which the memory decides if for an incoming data it is better to use \\(\\ell_2\\) or \\(\\ell_1\\) attentional bias:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.306,
+ 0.131,
+ 0.92,
+ 0.171
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = W _ {t - 1} - \\left\\{ \\begin{array}{l l} \\eta_ {t} \\nabla \\ell_ {2} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) & \\text {i f} \\| \\mathcal {M} (\\mathbf {k} _ {t}) - \\mathbf {v} _ {t} \\| \\leq \\delta_ {t}, \\\\ \\eta_ {t} \\delta_ {t} \\nabla \\ell_ {1} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) & \\text {O t h e r w i s e .} \\end{array} \\right. \\tag {16}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.179,
+ 0.423,
+ 0.194
+ ],
+ "angle": 0,
+ "content": "The role of parameter \\(\\delta_t\\) is the same as above."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.207,
+ 0.921,
+ 0.268
+ ],
+ "angle": 0,
+ "content": "Variant 3: Memory Robust to Value Shifts. Following the robustness requirement discussed in the previous section, we aim to design a memory mechanism that exhibits resilience against small shifts in the value parameter. A natural approach in this context is to employ a robust optimization formulation. Specifically, we define the loss function as the worst-case \\(\\ell_2\\) distance between the predicted memory output and the perturbed true value:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.321,
+ 0.276,
+ 0.92,
+ 0.307
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {L} \\left(\\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right); \\mathbf {v} _ {t}\\right) = \\max _ {\\| \\delta \\mathbf {v} _ {t} \\| _ {2} \\leq \\Delta} \\frac {1}{2} \\| \\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right) - \\left(\\mathbf {v} _ {t} + \\boldsymbol {\\delta} \\mathbf {v} _ {t}\\right) \\| _ {2} ^ {2}. \\tag {17}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.314,
+ 0.92,
+ 0.359
+ ],
+ "angle": 0,
+ "content": "This formulation seeks the memory parameters \\( W \\) that perform well even under the adverse local perturbation of the true value \\( \\mathbf{v}_t \\) within an \\( \\ell_2 \\) ball of radius \\( \\Delta \\). To solve the maximization problem in (17), we find the optimal perturbation \\( \\delta \\mathbf{v}_t^* \\). By solving this problem with respect to \\( \\delta \\mathbf{v}_t \\), we arrive at:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.423,
+ 0.366,
+ 0.609,
+ 0.4
+ ],
+ "angle": 0,
+ "content": "\\[\n\\delta \\mathbf {v} _ {t} ^ {*} = \\Delta \\frac {- \\mathcal {M} (W , \\mathbf {k} _ {t}) + \\mathbf {v} _ {t}}{\\| \\mathcal {M} (W , \\mathbf {k} _ {t}) - \\mathbf {v} _ {t} \\| _ {2}}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.406,
+ 0.757,
+ 0.421
+ ],
+ "angle": 0,
+ "content": "Substituting this optimal perturbation back into the loss function (17), we obtain the robust loss:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.281,
+ 0.429,
+ 0.75,
+ 0.457
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {L} \\left(\\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right); \\mathbf {v} _ {t}\\right) = \\frac {1}{2} \\| \\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2} ^ {2} + \\Delta \\| \\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2} + \\frac {1}{2} \\Delta^ {2}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.464,
+ 0.92,
+ 0.51
+ ],
+ "angle": 0,
+ "content": "This robust loss function is a combination of the standard \\(\\ell_2\\) loss and a term proportional to the \\(\\ell_2\\) norm of the error, scaled by the robustness parameter \\(\\Delta\\). The value of \\(\\Delta\\) thus controls the trade-off between fitting the nominal data and ensuring robustness against value perturbations."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.517,
+ 0.92,
+ 0.577
+ ],
+ "angle": 0,
+ "content": "For simplicity of the derivations, let us consider a constant value for \\(\\Delta\\), an Euclidean retention gate \\(\\mathrm{Ret}_t(W,W_{t - 1}) = \\| W - W_{t - 1}\\|^2\\), and an attentional bias term \\(\\widetilde{\\ell} (W;\\mathbf{k}_t,\\mathbf{v}_t) = \\langle W - W_{t - 1},\\nabla \\ell (W_{t - 1};\\mathbf{k}_t,\\mathbf{v}_t)\\rangle\\). Furthermore, to simplify the memory operation, we assume a linear matrix memory model \\(\\mathcal{M}(W,\\mathbf{k}_t) = W\\mathbf{k}_t\\). Under these assumptions, we can derive the memory update mechanism using gradient descent on the robust loss:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.289,
+ 0.595,
+ 0.741,
+ 0.629
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = W _ {t - 1} - \\eta \\left(\\left(\\mathcal {M} \\left(W _ {t - 1}, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {\\top} + \\Delta \\frac {\\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}}{\\| \\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2}} \\mathbf {k} _ {t} ^ {\\top}\\right)\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.632,
+ 0.919,
+ 0.663
+ ],
+ "angle": 0,
+ "content": "In this update rule, the parameter \\(\\Delta\\), which governs the influence of the robustness term, can also be treated as a learnable parameter, allowing the model to adapt its robustness based on the observed data."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.113,
+ 0.68,
+ 0.41,
+ 0.696
+ ],
+ "angle": 0,
+ "content": "5.2 Alternative Retention Gates"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.711,
+ 0.92,
+ 0.77
+ ],
+ "angle": 0,
+ "content": "Variant 1: Memorization Over A Scaled Probability Simplex Via \\( f \\)-Divergence. A common technique in learning to prevent numerical instabilities and exploding values is to restrict the search space to a bounded domain. Following this principle, to avoid numerical instabilities, we can constrained the variable \\( W_{t} \\) to lie within a (scaled) probability simplex. In other words, we can restrict the state to lie in the constraint set"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.373,
+ 0.78,
+ 0.658,
+ 0.797
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {W} = \\{W \\mid \\| W \\| _ {1} = c \\text {a n d} W _ {j l} \\geq 0, \\forall j, l \\}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.806,
+ 0.922,
+ 0.867
+ ],
+ "angle": 0,
+ "content": "In this set, each matrix \\( W \\) can be viewed as a measure. Thus, in (Learning-Retaining Viewpoint), we can utilize divergences over measures to define our premetric. For example, we can use \\( f \\)-divergence measure (Polyanskiy et al. 2025, Def 4.9), (Csiszar 1967) to define \\( \\mathrm{D}_t(\\cdot, \\cdot) \\). More specifically, let \\( f(\\cdot) \\) be a smooth strictly convex function from \\( \\mathbb{R}^+ \\) to \\( \\mathbb{R} \\) with \\( f(1) = 0 \\). Then, we can define the \\( f \\)-divergence between \\( W \\) and \\( W' \\) as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.411,
+ 0.874,
+ 0.619,
+ 0.915
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathrm {D} _ {t} (W, W ^ {\\prime}) = \\sum_ {j l} W _ {j l} ^ {\\prime} f \\left(\\frac {W _ {j l}}{W _ {j l} ^ {\\prime}}\\right).\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.509,
+ 0.938,
+ 0.526,
+ 0.949
+ ],
+ "angle": 0,
+ "content": "10"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.093,
+ 0.921,
+ 0.137
+ ],
+ "angle": 0,
+ "content": "It is known that \\( f \\)-divergence is zero if and only if \\( W = W' \\); see Polyanskiy et al. 2025, Theorem 2.3. Using the above premetric as the retention gate and setting \\( \\widetilde{\\ell}(W; \\mathbf{k}_t, \\mathbf{v}_t) = \\langle W - W_{t-1}, \\nabla \\ell(W_{t-1}; \\mathbf{k}_t, \\mathbf{v}_t) \\rangle \\) in (Learning-Retaining Viewpoint), we get the update rule"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.375,
+ 0.139,
+ 0.92,
+ 0.154
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = W _ {t - 1} \\odot g \\left(- \\zeta_ {t} - \\eta_ {t} \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right). \\tag {18}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.161,
+ 0.92,
+ 0.267
+ ],
+ "angle": 0,
+ "content": "Here \\( g(\\cdot) \\) is the inverse of the mapping \\( f' \\), i.e., \\( g(f'(\\tau)) = \\tau \\), \\( \\forall \\tau \\); the operator \\( \\odot \\) denotes the Hadamard (elementwise) product, and \\( \\zeta_t \\) should be chosen such that \\( \\| W_t\\|_1 = c \\). Notice that since the function \\( f(\\cdot) \\) is strictly convex and smooth, its derivative is strictly increasing and hence \\( g(\\cdot) \\) is well defined. Conversely, for any strictly monotone function \\( g(\\cdot) \\), we can find its inverse function \\( g^{-1} \\) (which is strictly increasing) and define \\( f(\\tau) = \\mathrm{const} + \\int_{\\tau' = 0}^{\\infty}g^{-1}(\\tau')d\\tau' \\). The term const should be chosen such that \\( f(1) = 0 \\). Then the update rule in (18) can be interpreted by the \\( f \\)-divergence regularization, as explained above. Therefore, one can directly choose a continuous monotonically increasing function \\( g(\\cdot) \\) and use (18) for memory update."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.281,
+ 0.921,
+ 0.363
+ ],
+ "angle": 0,
+ "content": "Specializing to KL divergence. Let us further make the above update rule explicit by using special function \\( f \\). If we choose \\( f(\\tau) = \\tau \\ln(\\tau) \\), then the \\( f \\)-divergence becomes the widely used KL divergence measure \\( D_t(W, W_{t-1}) = \\sum_{jl} W_{jl} \\log \\left( \\frac{W_{jl}}{(W_t)_{jl}} \\right) \\). In addition, we can also utilize the Shannon entropy as the global retention by regularizing deviations from uniform distribution, i.e., \\( G_t(W) = \\sum_{jl} W_{jl} \\log (W_{jl}) \\). Combining these choices of the local and global retention gates, we obtain the overall retention gate"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.307,
+ 0.363,
+ 0.724,
+ 0.401
+ ],
+ "angle": 0,
+ "content": "\\[\n\\operatorname {R e t} _ {t} (W, W _ {t - 1}) = \\frac {1}{\\eta_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(\\frac {W _ {j l}}{\\left(W _ {t}\\right) _ {j l}}\\right) + \\frac {1}{\\alpha_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(W _ {j l}\\right)\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.408,
+ 0.919,
+ 0.44
+ ],
+ "angle": 0,
+ "content": "Choosing the attentional bias \\(\\widetilde{\\ell}(W; \\mathbf{k}_t, \\mathbf{v}_t) = \\langle W - W_{t-1}, \\nabla \\ell(W_{t-1}; \\mathbf{k}_t, \\mathbf{v}_t) \\rangle\\) and the above retention gate will lead to the update rule"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.216,
+ 0.45,
+ 0.92,
+ 0.488
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = \\arg \\min _ {W} \\left\\langle W - W _ {t - 1}, \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) \\right\\rangle + \\frac {1}{\\eta_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(\\frac {W _ {j l}}{\\left(W _ {t}\\right) _ {j l}}\\right) + \\frac {1}{\\alpha_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(W _ {j l}\\right) \\tag {19}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.273,
+ 0.491,
+ 0.92,
+ 0.525
+ ],
+ "angle": 0,
+ "content": "\\[\n\\text {s . t .} \\quad \\sum_ {j l} W _ {j l} = c, W _ {j l} \\geq 0, \\forall j l \\tag {20}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.536,
+ 0.68,
+ 0.551
+ ],
+ "angle": 0,
+ "content": "Attaching the Lagrange multiplier to the first constraint, the KKT conditions imply"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.247,
+ 0.562,
+ 0.787,
+ 0.597
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left(\\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right) _ {j l} + \\left(\\frac {1}{\\eta_ {t}} + \\frac {1}{\\alpha_ {t}}\\right) \\left(1 + \\log W _ {j l}\\right) - \\frac {1}{\\eta_ {t}} \\log \\left(\\left(W _ {t - 1}\\right) _ {j l}\\right) + \\mu_ {t} = 0, \\quad \\forall j, l\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.608,
+ 0.919,
+ 0.64
+ ],
+ "angle": 0,
+ "content": "where \\(\\mu_t\\) should be chosen such that \\(\\sum_{jl} W_{jl} = c\\). Rearranging the terms and defining \\(\\lambda_t = \\frac{1 / \\alpha_t}{1 / \\alpha_t + 1 / \\eta_t}\\), \\(\\eta_t' = \\frac{1}{1 / \\alpha_t + 1 / \\eta_t}\\), we get the update rule"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.322,
+ 0.641,
+ 0.92,
+ 0.657
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} \\leftarrow c \\operatorname {S o f t m a x} \\left(\\left(1 - \\lambda_ {t}\\right) \\log \\left(W _ {t - 1}\\right) - \\eta_ {t} ^ {\\prime} \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right) \\tag {21}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.664,
+ 0.92,
+ 0.694
+ ],
+ "angle": 0,
+ "content": "where \\(\\lambda_t \\in (0,1)\\) and \\(\\eta' \\in \\mathbb{R}^+\\) are the parameters that can be learned during training. The Softmax operator ensures that the output lies in the set \\(\\mathcal{W}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.702,
+ 0.919,
+ 0.732
+ ],
+ "angle": 0,
+ "content": "Notice that while all above calculations are done for a matrix \\( W \\), similar update rule holds for other forms of parameters such as when \\( W \\) is a neural network (or when the parameter \\( W \\) is normalized per slice)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.745,
+ 0.92,
+ 0.821
+ ],
+ "angle": 0,
+ "content": "Variant 2: Elastic Net Regularization: Hard and Soft Forgetting. Elastic net is a powerful and popular tool in regression analysis to balance the feature selection capabilities of LASSO (Tibshirani 1996) and bias reduction properties of Ridge regression (Hilt et al. 1977; Hoerl et al. 1970). It has been widely used in different applications due to its ability to handle high-dimensional data and mitigate the effects of multicollinearity. Given this success, a natural question is what happens if we use this regularization scheme in our context."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.829,
+ 0.933,
+ 0.874
+ ],
+ "angle": 0,
+ "content": "Let us start based on (Learning-Retaining Viewpoint) to design our memorization scheme. In (Learning-Retaining Viewpoint), we discussed that the loss function \\(\\widetilde{\\ell_t} (W;\\mathbf{k}_t,\\mathbf{v}_t)\\) is an approximation of the original function \\(\\ell (\\cdot)\\), measuring our goodness-of-fit. Regularizing this loss with elastic net regularizer, we obtain the approximation"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.366,
+ 0.884,
+ 0.666,
+ 0.902
+ ],
+ "angle": 0,
+ "content": "\\[\n\\widetilde {\\ell} _ {t} (W; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) = \\langle W - W _ {t - 1}, \\nabla \\ell (W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) \\rangle .\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.509,
+ 0.938,
+ 0.525,
+ 0.949
+ ],
+ "angle": 0,
+ "content": "11"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.092,
+ 0.921,
+ 0.137
+ ],
+ "angle": 0,
+ "content": "with a global retention of \\( \\mathrm{G}_t(W) = \\frac{1}{2\\beta} \\| W\\| _2^2 +\\frac{1}{\\alpha}\\| W\\| _1 \\). To fully specify the update rule of (Learning-Retaining Viewpoint), we also need to specify the premetric functions \\( \\mathrm{D}_t(\\cdot ,\\cdot) \\). For the sake of keeping the update rule simple (and parallelizable), we can choose"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.41,
+ 0.136,
+ 0.622,
+ 0.164
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathrm {D} _ {t} (W, W _ {t - 1}) = \\frac {1}{2} \\| W - W _ {t - 1} \\| _ {2} ^ {2}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.168,
+ 0.717,
+ 0.184
+ ],
+ "angle": 0,
+ "content": "These choices of the attentional bias and retention gate leads to the following update rule:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.39,
+ 0.206,
+ 0.92,
+ 0.222
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = \\mathcal {S} _ {Y} \\left(\\lambda W _ {t - 1} - \\zeta \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right), \\tag {22}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.228,
+ 0.919,
+ 0.262
+ ],
+ "angle": 0,
+ "content": "where \\(\\gamma = \\frac{\\eta\\beta}{\\alpha(\\eta + \\beta)}\\), \\(\\lambda = \\frac{\\beta}{\\beta + \\eta}\\), \\(\\zeta = \\eta\\lambda\\), and \\(S_{\\gamma}\\) is the soft thresholding operator, applied element-wise. For each element, this operator is defined as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.404,
+ 0.263,
+ 0.626,
+ 0.28
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {S} _ {\\gamma} (z) = \\operatorname {s i g n} (z) \\max \\left\\{0, | z | - \\gamma \\right\\}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.285,
+ 0.921,
+ 0.317
+ ],
+ "angle": 0,
+ "content": "In other words, for large values of \\( z \\), \\( S_{\\gamma}(z) \\) makes \\( z \\) closer to zero by \\( \\gamma \\) amount. If it is already in the \\( \\gamma \\)-vicinity of zero, then it makes it zero (hard forget)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.323,
+ 0.92,
+ 0.391
+ ],
+ "angle": 0,
+ "content": "Equation (22) can be viewed as a combination of soft forgetting (obtained by multiplying \\( W \\) by \\( \\lambda \\in (0,1) \\), and a hard forgetting (if it is smaller than \\( \\gamma \\)). The hyperparameters \\( \\gamma, \\lambda, \\) and \\( \\zeta \\) can be learned. Notice that since the shrinkage operator is not differentiable, we can approximate it with its smooth approximation. For example, we can use \\( S_{\\gamma}(z) \\approx \\frac{|z|*\\arctan(z / \\gamma)}{\\pi / 2} \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.402,
+ 0.92,
+ 0.432
+ ],
+ "angle": 0,
+ "content": "Variant 3: Elastic Net Regularization: Forgetting via Soft-thresholding. The elastic net regularizer can also be used in the (FTRL Viewpoint). In particular, in (FTRL Viewpoint), we can set"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.415,
+ 0.442,
+ 0.617,
+ 0.474
+ ],
+ "angle": 0,
+ "content": "\\[\n\\frac {1}{\\eta_ {t}} R _ {t} (W) = \\frac {1}{\\eta} \\| W \\| ^ {2} + \\frac {1}{\\alpha} \\| W \\| _ {1}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.485,
+ 0.92,
+ 0.517
+ ],
+ "angle": 0,
+ "content": "and use \\(\\widehat{\\ell}(W; x_i) = \\langle W - W_{i-1}, \\nabla \\ell(W_{i-1}; x_i) \\rangle\\). Assuming initialization at \\(W_0 = 0\\), these choices of attentional bias and retention gate leads to the update rules:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.414,
+ 0.529,
+ 0.616,
+ 0.544
+ ],
+ "angle": 0,
+ "content": "\\[\nA _ {t} = A _ {t - 1} - \\eta \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.416,
+ 0.547,
+ 0.918,
+ 0.565
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = \\mathcal {S} _ {\\eta / \\alpha} (A _ {t}) \\tag {23}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.575,
+ 0.871,
+ 0.591
+ ],
+ "angle": 0,
+ "content": "Here \\(S_{\\eta /\\alpha}(\\cdot)\\) is the soft-thresholding operator with parameter \\(\\eta /\\alpha\\) , which can be smoothly as explained in Variant 1.1."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.603,
+ 0.919,
+ 0.651
+ ],
+ "angle": 0,
+ "content": "Variant 4: General \\( L_{q} \\) Memory Stability. Existing work is based on the retention gate choices \\( \\mathrm{D}_t(W, W_{t-1}) = \\|W - W_{t-1}\\|_F^2 \\) or \\( R(W) = \\|W\\|_2^2 \\). However, one can choose other choices of retention gate. For example, in (FTRL Viewpoint), we can choose \\( L_{q} \\) norm as the regularizer \\( R(W) \\). More specifically, for \\( 1 < q \\leq 2 \\), we can set"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.422,
+ 0.66,
+ 0.609,
+ 0.692
+ ],
+ "angle": 0,
+ "content": "\\[\n\\frac {1}{\\eta_ {t}} R (W) = \\frac {1}{2 \\eta (q - 1)} \\| W \\| _ {q} ^ {2}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.703,
+ 0.92,
+ 0.772
+ ],
+ "angle": 0,
+ "content": "Using this retention gate and choosing \\(\\widehat{\\ell_i} (W;\\mathbf{k}_t,\\mathbf{v}_t) = \\langle W - W_{i - 1},\\nabla \\ell (W_{i - 1};\\mathbf{k}_t,\\mathbf{v}_t)\\rangle\\) in (FTRL Viewpoint), leads to the update rule \\(W_{t} = -\\eta \\frac{A_{t}}{\\|A_{t}\\|_{p}^{p - 2}}\\), where \\(p = \\frac{q}{q - 1}\\) and \\(A_{t} = \\sum_{i = 1}^{t}\\nabla \\ell (W_{i - 1};\\mathbf{k}_{t},\\mathbf{v}_{t})\\); see Shalev-Shwartz et al. 2012, Section 2.6. Here, \\(\\odot\\) denotes the Hadamard (element-wise) product and \\(|\\cdot |\\) is the element-wise absolute value operator. Assuming \\(W_0 = 0\\), this update rule can be recursively written as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.333,
+ 0.781,
+ 0.698,
+ 0.817
+ ],
+ "angle": 0,
+ "content": "\\[\nA _ {t} = A _ {t - 1} - \\eta \\nabla \\ell \\left(W _ {i - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right), \\quad \\text {a n d} \\quad W _ {t} = \\frac {A _ {t}}{\\| A _ {t} \\| _ {p} ^ {p - 2}}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.841,
+ 0.92,
+ 0.887
+ ],
+ "angle": 0,
+ "content": "Variant 5: Bregman Divergence as Retention Gate.. Another natural choice is to use Bregman divergence as retention gate, leading to a mirror descent-type algorithms. In particular, given a smooth strictly convex function \\( f(\\cdot): \\mathbb{R} \\mapsto \\mathbb{R} \\), we can define the function \\( F(W) = \\sum_{jl} f(W_{jl}) \\). Based on this choice of function \\( F \\), we define the Bregman divergence"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.362,
+ 0.898,
+ 0.667,
+ 0.914
+ ],
+ "angle": 0,
+ "content": "\\[\nD _ {t} (W, W ^ {\\prime}) = F (W) - F \\left(W ^ {\\prime}\\right) - \\langle W ^ {\\prime}, W - W ^ {\\prime} \\rangle\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.509,
+ 0.938,
+ 0.526,
+ 0.948
+ ],
+ "angle": 0,
+ "content": "12"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.092,
+ 0.92,
+ 0.123
+ ],
+ "angle": 0,
+ "content": "as our parametric function. Utilizing this retention gate and choosing \\(\\widetilde{\\ell}_t(W;\\mathbf{k}_t,\\mathbf{v}_t) = \\langle W - W_{t - 1},\\nabla \\ell (W_{t - 1};\\mathbf{k}_t,\\mathbf{v}_t)\\rangle\\) in (Learning-Retaining Viewpoint), we obtain the update rule"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.38,
+ 0.133,
+ 0.652,
+ 0.15
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = g \\left(- \\eta \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) + F ^ {\\prime} \\left(W _ {t - 1}\\right)\\right).\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.159,
+ 0.92,
+ 0.19
+ ],
+ "angle": 0,
+ "content": "Here, \\( F' \\) is the mapping obtained by applying \\( f'(\\cdot) \\) (the derivative of \\( f \\)) element-wise to all entries of its input matrix argument. The function \\( g \\) is the inverse of the mapping \\( F'(\\cdot) \\), i.e., \\( g(F'(W)) = W \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.198,
+ 0.922,
+ 0.308
+ ],
+ "angle": 0,
+ "content": "If we choose \\( f(\\tau) = \\frac{\\tau^2}{2} \\), then \\( F'(W) \\) becomes the identity mapping and so is \\( g \\). Therefore, the above update becomes simple gradient descent with no nonlinearity involved in the update rule. However, other choices of \\( f(\\cdot) \\) introduces additional nonlinearity in \\( g(\\cdot) \\), which can enhance the expressivity of our memory. For example, we can choose the function \\( f(\\cdot) \\) so that its derivative becomes the inverse sigmoid function, i.e., \\( f'(\\tau) = \\ln \\left( \\frac{\\tau}{1 - \\tau} \\right) \\) with \\( f': (0,1) \\mapsto \\mathbb{R} \\). Since \\( f'(\\cdot) \\) is strictly increasing, then the function \\( f(\\cdot) \\) (and hence \\( F(\\cdot) \\)) is strictly convex. Therefore, the Bregman divergence is well defined. Moreover, the inverse of the function \\( f'(\\cdot) \\) becomes the sigmoid function, i.e., \\( g(\\tau) = \\sigma(\\tau) = \\frac{\\exp(\\tau)}{1 + \\exp(\\tau)} \\) with \\( g: \\mathbb{R} \\mapsto (0,1) \\). Then, the update of the memory becomes"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.373,
+ 0.317,
+ 0.66,
+ 0.352
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = \\sigma \\left(\\ln \\left(\\frac {W _ {t}}{1 - W _ {t}}\\right) - \\eta \\nabla \\ell (W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t})\\right),\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.361,
+ 0.92,
+ 0.392
+ ],
+ "angle": 0,
+ "content": "where \\(\\sigma\\) is the sigmoid function operated element-wise on the entries of \\(W\\), and the division operator \\(\\frac{W_t}{1 - W_t}\\) is also performed element-wise. This update rule guarantees that the elements of \\(W_t\\) remain within the interval \\((0, 1)\\)."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.113,
+ 0.41,
+ 0.58,
+ 0.427
+ ],
+ "angle": 0,
+ "content": "5.3 MIRAs's Variants: MONETA, YAAD, and MEMORA"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.435,
+ 0.92,
+ 0.482
+ ],
+ "angle": 0,
+ "content": "In the previous section we discussed different potential choices for attentional bias and retention gate to show the generality and the potential of MIRAs. In this section, building upon our framework, we present three novel sequence models, each of which designed based on a different motivation, and discuss how they can leverage fast parallel training."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.493,
+ 0.92,
+ 0.587
+ ],
+ "angle": 0,
+ "content": "MOnETA. Given \\(p,q\\in \\mathbb{R}^{\\geq 1}\\), we design \\((p,q)\\)-MONETA as the variant of MIRAs as follows: (1) For the choice of memory architecture, we use an MLP with 2 layers with expansion factor of 4 and GELU activation function (Hendrycks et al. 2016). We also use residual connections and layer norm, resulting in \\(\\mathcal{M}(x) = x + \\mathsf{LN}(W_1\\sigma (W_2x))\\). (2) We choose \\(\\ell_p\\)-attentional bias (introduced in Equation 11) for MONETA. (3) For the choice of retention gate, we use the hybrid of \\(\\ell_q\\) retention gate \\(\\frac{1}{2(q - 1)}\\| W\\| _q^2\\) (see Section 5.2 for details) and the standard \\(\\ell_2\\) regularization \\(\\frac{1}{\\beta}\\| W\\| _2^2\\). (4) Finally, we use gradient descent as the memory learning algorithm. The above choices, result in the following recurrent formula for the memory module:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.32,
+ 0.594,
+ 0.92,
+ 0.631
+ ],
+ "angle": 0,
+ "content": "\\[\nA _ {t} = \\alpha_ {t} A _ {t - 1} - \\eta_ {t} \\nabla \\ell_ {p} \\left(W _ {i - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right), \\quad \\text {a n d} \\quad W _ {t} = \\frac {A _ {t}}{\\| A _ {t} \\| _ {q} ^ {q - 2}}. \\tag {24}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.639,
+ 0.421,
+ 0.654
+ ],
+ "angle": 0,
+ "content": "Notably the gradient can be calculated using:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.305,
+ 0.663,
+ 0.92,
+ 0.681
+ ],
+ "angle": 0,
+ "content": "\\[\n\\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) = p \\eta_ {t} \\left(\\operatorname {S i g n} \\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\odot | W \\mathbf {k} _ {t} - \\mathbf {v} _ {t} | ^ {p - 1}\\right) \\mathbf {k} _ {t} ^ {\\top}. \\tag {25}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.69,
+ 0.263,
+ 0.707
+ ],
+ "angle": 0,
+ "content": "We use \\((p,q) = (3,4)\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.718,
+ 0.92,
+ 0.826
+ ],
+ "angle": 0,
+ "content": "YAAD. Building upon our discussion on the importance of robust memory that protects itself from extreme events (tokens), we design YAAD based on Huber objective. That is, in MirAS, for the choice of memory structure, we follow MONETA and use an MLP with the same architecture as above; for the choice of attentional bias, we use Huber loss (defined in Equation 16); for the choice retention gate, for the sake of simplicity, we use a combination of local and global retention as \\(\\mathrm{Ret}_t(W,W_{t - 1}) = \\frac{1}{2\\theta_t}\\| W - W_{t - 1}\\| _F^2 +\\frac{1}{\\beta_t}\\| W\\| _2^2\\) , which is equivalent to the \"forget gate\" mechanism introduced by Behrouz et al. (2024c); and finally, we simply use gradient descent as the memory learning algorithm. Given the above choices, we can write the resulted memory learning process as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.299,
+ 0.835,
+ 0.92,
+ 0.876
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = \\alpha_ {t} W _ {t - 1} - \\left\\{ \\begin{array}{l l} \\eta_ {t} \\nabla \\ell_ {2} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) & \\text {i f} \\| \\mathcal {M} (\\mathbf {k} _ {t}) - \\mathbf {v} _ {t} \\| \\leq \\delta_ {t}, \\\\ \\eta_ {t} \\delta_ {t} \\nabla \\ell_ {1} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) & \\text {O t h e r w i s e .} \\end{array} \\right. \\tag {26}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.883,
+ 0.92,
+ 0.915
+ ],
+ "angle": 0,
+ "content": "Note that for improving the expressive power, in all architectures, we decouple the learning rate \\(\\eta\\) and the retention gate rate \\(\\alpha\\), resulting in an independent parameter \\(\\beta_{t} \\in [0,1]^{d}\\)."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.509,
+ 0.938,
+ 0.526,
+ 0.949
+ ],
+ "angle": 0,
+ "content": "13"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.182,
+ 0.09,
+ 0.345,
+ 0.351
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.367,
+ 0.089,
+ 0.532,
+ 0.35
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.559,
+ 0.09,
+ 0.852,
+ 0.35
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.111,
+ 0.366,
+ 0.92,
+ 0.398
+ ],
+ "angle": 0,
+ "content": "Figure 2: Visualization of the MirAs's variant architecture, their hybrid counterpart with SWA, and block design of MirAs layer."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.424,
+ 0.92,
+ 0.5
+ ],
+ "angle": 0,
+ "content": "MEMORA. Finally, in MEMORA, we use the idea of elastic net regularization (i.e., hard and soft retention). To this end, in Miras: (1) For the choice of memory architecture, similar to above variants, we use an MLP (the same architecture as the previous variants). (2) For the choice of attentional bias, we use simple \\(\\ell_2\\) regression loss. (3) For the choice of retention gate we use KL divergence as in Equation 21. (4) Finally, we optimize the memory using gradient descent, resulting in the following update rule:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.347,
+ 0.534,
+ 0.92,
+ 0.551
+ ],
+ "angle": 0,
+ "content": "\\[\nW _ {t} = \\operatorname {S o f t m a x} \\left(\\alpha_ {t} \\log \\left(W _ {t - 1}\\right) - \\eta_ {t} \\nabla \\ell_ {2} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right) \\tag {27}\n\\]"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.113,
+ 0.581,
+ 0.529,
+ 0.599
+ ],
+ "angle": 0,
+ "content": "5.4 Architecture Backbone and Fast Training"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.611,
+ 0.92,
+ 0.718
+ ],
+ "angle": 0,
+ "content": "Architectural Backbone. For the architectural backbone, we fully follow recent studies (Behrouz et al. 2024c; Yang et al. 2024a): We replace attention modules with our variants of MIRAs in Llama's macro architecture with MLPs with SwiGLU(. ) activation, rotary positional encodings (RoPE) (Su et al. 2024), and RMSNorm (Zhang et al. 2019). For MIRAs layer block, we follow the recent modern linear recurrent models (Behrouz et al. 2024c; Yang et al. 2024a), and incorporate a 1D depthwise-separable convolution layer (with kernel size of 4) after each of the query, key, and value projections. For the sake of training stability, we also use \\(\\ell_2\\) normalization to \\(\\mathbf{q}\\) and \\(\\mathbf{k}\\). The output of MIRAs layer block is normalized and gated with a linear layer (Mehta et al. 2023)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.731,
+ 0.92,
+ 0.792
+ ],
+ "angle": 0,
+ "content": "Channel-wise Parameters. For learnable parameters of \\(\\eta_t, \\delta_t\\) and the retention gate of \\(\\alpha_t\\) we use channel-wise parametrization, i.e., \\(\\eta_t, \\delta_t, \\alpha_t \\in \\mathbb{R}^d\\). While gaining more expressive power, this parametrization results in significant parameter increase. To mitigate this issue, following Peng et al. (2025b), we use low-rank projections to project the input into \\(\\mathbb{R}^k\\) and then to \\(\\mathbb{R}^d\\), where \\(k\\) is a hyperparameter (usually 32 or 64). The backbone architecture is illustrated in Figure 2."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.805,
+ 0.919,
+ 0.851
+ ],
+ "angle": 0,
+ "content": "Hybrid Models. We also evaluate the hybrid version of Miras's variants. For hybrid models, we follow the Samba (Ren et al. 2024) architecture, in which we sequentially combine our Miras layer with Sliding Window Attention (SWA). The illustration of hybrid model Figure 2."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.863,
+ 0.92,
+ 0.91
+ ],
+ "angle": 0,
+ "content": "Parallelizable Training. While the design of Miras's variant are theoretically well-motivated, their recurrence is non-linear, potentially making their straightforward training slow for large scales. In this section, we build upon the work of Behrouz et al. (2024c) and Sun et al. (2024) to make the training parallelizable. The main idea is to divide the sequence into"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.509,
+ 0.938,
+ 0.526,
+ 0.949
+ ],
+ "angle": 0,
+ "content": "14"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.093,
+ 0.92,
+ 0.138
+ ],
+ "angle": 0,
+ "content": "chunks with size \\( b \\) (usually is 16 or 64) and calculate the gradient for all tokens in the current chunk with respect to the last state of the memory in the previous chunk. That is, we use \\( \\nabla \\ell(\\mathcal{M}_{t'}; \\mathbf{k}_t, \\mathbf{v}_t) \\) instead of \\( \\nabla \\ell(\\mathcal{M}_{t-1}; \\mathbf{k}_t, \\mathbf{v}_t) \\), where \\( t' \\) is the last state in the previous chunk."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.145,
+ 0.92,
+ 0.207
+ ],
+ "angle": 0,
+ "content": "Given the above trick, we can calculate all gradients at once and make the recurrence inside each chunk linear. However, to fully take advantage of accelerators, we need to reformulate the process as matrix multiplication. For MONETA, for the sake of clarity, assume \\( q = 2 \\). We follow the same algorithm as Behrouz et al. (2024c) and expand the recurrence as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.389,
+ 0.219,
+ 0.92,
+ 0.276
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\mathcal {M} _ {t} = \\alpha_ {t} \\mathcal {M} _ {t - 1} - \\eta_ {t} \\nabla \\ell (\\mathcal {M} _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) \\\\ = \\beta_ {t} \\mathcal {M} _ {0} - \\sum_ {i = 1} ^ {t} \\eta_ {i} \\frac {\\beta_ {t}}{\\beta_ {i}} \\nabla \\ell \\left(\\mathcal {M} _ {t ^ {\\prime}}; \\mathbf {k} _ {i}, \\mathbf {v} _ {i}\\right), \\tag {28} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.288,
+ 0.92,
+ 0.334
+ ],
+ "angle": 0,
+ "content": "where \\(t' = t - \\mathrm{mod}(t, b)\\), and \\(\\beta_{i} = \\prod_{j=1}^{i} \\alpha_{j}\\). For the sake of clarity, we focus on the first chunk, i.e., \\(t = b\\) and so \\(t' = 0\\), and explain the process for the case that \\(\\mathcal{M}_t = W_t\\) is linear. The process for 2-layer MLPs and other chunks is similar. Using \\(\\ell_p\\) loss function, we have:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.255,
+ 0.345,
+ 0.92,
+ 0.405
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\nabla \\ell \\left(W _ {0}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) = p \\left(\\operatorname {S i g n} \\left(W _ {0} \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\odot \\left| W _ {0} \\mathbf {k} _ {t} - \\mathbf {v} _ {t} \\right| ^ {p - 1}\\right) \\mathbf {k} _ {t} ^ {\\top} \\\\ \\Rightarrow \\sum_ {i = 1} ^ {b} \\eta_ {i} \\frac {\\beta_ {b}}{\\beta_ {i}} \\nabla \\ell \\left(W _ {0};; \\mathbf {k} _ {i}, \\mathbf {v} _ {i}\\right) = p \\mathbf {E} _ {b} \\odot \\mathbf {B} _ {b} \\odot \\operatorname {S i g n} \\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\odot \\left(\\left| W _ {0} \\mathbf {K} - \\mathbf {V} \\right| ^ {p - 1}\\right) \\mathbf {K} ^ {\\top}, \\tag {29} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.418,
+ 0.92,
+ 0.485
+ ],
+ "angle": 0,
+ "content": "where \\(\\mathbf{E}_b = \\left[\\eta_1\\quad \\eta_2\\quad \\dots \\quad \\eta_b\\right]\\) and \\(\\mathbf{B}_b\\) is defined analogously on \\(\\frac{\\beta_b}{\\beta_i}\\mathrm{s}\\). For the sake of stability in training, we use \\(\\operatorname{Sign}(x)\\approx \\tanh (\\alpha x)\\) and \\(|x| = \\sqrt{x^2 + \\epsilon}\\), where \\(\\epsilon >0\\) is a small number (i.e., \\(\\epsilon = 1e - 6\\)). As discussed in Equation 24, the case that \\(q\\neq 2\\) appears as a normalization term on the memory. Similar to Titans (Behrouz et al. 2024c) and TTT (Sun et al. 2024), we do not apply this non-linearity inside each chunk and instead use it at the end of each chunk."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.492,
+ 0.919,
+ 0.524
+ ],
+ "angle": 0,
+ "content": "For YAAD, the process is very similar to the above. We calculate the gradient of both \\(\\ell_1\\) and \\(\\ell_2\\) loss and use a masking based on \\(\\| \\mathcal{M}(\\mathbf{k}_t) - \\mathbf{v}_t\\| \\leq \\delta_t\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.53,
+ 0.92,
+ 0.605
+ ],
+ "angle": 0,
+ "content": "For MEMORA, the update rule has two non-linear parts, i.e., softmax and log, making the model hardly parallelizable. To this end, as discussed above, we use its linear version inside each chunk and its non-linear version across chunks. However, using both log and softmax at the end of each chunk removes the effect of log. To this end, we consider a lag tokens after each chunk (i.e., tokens with index \\( i = kb + 1 \\), where \\( b \\) is the chunk size and \\( k \\in \\mathbb{Z}^+ \\)). That is, let \\( \\mathcal{M}_0 \\) be the last state of the memory in previous chunk, we have:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.352,
+ 0.618,
+ 0.92,
+ 0.633
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {M} _ {1} = \\operatorname {S o f t m a x} \\left(\\alpha_ {1} \\log \\left(\\mathcal {M} _ {0}\\right) - \\eta_ {1} \\nabla \\ell_ {2} \\left(\\mathcal {M} _ {0}; \\mathbf {k} _ {1}, \\mathbf {v} _ {1}\\right)\\right), \\tag {30}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.644,
+ 0.871,
+ 0.66
+ ],
+ "angle": 0,
+ "content": "and then we use \\(\\mathcal{M}_1\\) for the next chunk. Again, for the sake of clarity, assume that memory is linear, i.e., \\(\\mathcal{M}_1 = W_1\\):"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.336,
+ 0.671,
+ 0.92,
+ 0.73
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\nabla \\ell \\left(W _ {1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) = \\left(W _ {1} \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {\\top} (31) \\\\ \\Rightarrow \\sum_ {i = 1} ^ {b} \\eta_ {i} \\frac {\\beta_ {b}}{\\beta_ {i}} \\nabla \\ell \\left(W _ {1};; \\mathbf {k} _ {i}, \\mathbf {v} _ {i}\\right) = \\mathbf {E} _ {b} \\odot \\mathbf {B} _ {b} \\odot \\left(W _ {1} \\mathbf {K} - \\mathbf {V}\\right) \\mathbf {K} ^ {\\top}, (32) \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.741,
+ 0.488,
+ 0.756
+ ],
+ "angle": 0,
+ "content": "where matrices are defined the same as for Equation 29."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.113,
+ 0.779,
+ 0.291,
+ 0.799
+ ],
+ "angle": 0,
+ "content": "6 Experiments"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.809,
+ 0.919,
+ 0.87
+ ],
+ "angle": 0,
+ "content": "In our experimental evaluations, we aim to answer three main questions: (1) Does different attentional biases results in different architectures in practice? (2) How does different types of retention gates (i.e., retention gate) affect the performance of the model in long context? (3) How do MEMORA, MONETA, and YAAD perform in downstream tasks compare to baselines?"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.883,
+ 0.92,
+ 0.914
+ ],
+ "angle": 0,
+ "content": "Setup. We train our models with training context window of size 4096 using either FineWeb-Edu dataset (Penedo et al. 2024) (for LM and common-sense reasoning tasks) or C4 dataset (Raffel et al. 2020) (for scaling patterns). We use model"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.509,
+ 0.938,
+ 0.526,
+ 0.949
+ ],
+ "angle": 0,
+ "content": "15"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.118,
+ 0.092,
+ 0.378,
+ 0.216
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.389,
+ 0.092,
+ 0.647,
+ 0.216
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.66,
+ 0.092,
+ 0.918,
+ 0.216
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.111,
+ 0.228,
+ 0.921,
+ 0.259
+ ],
+ "angle": 0,
+ "content": "Figure 3: Scaling patterns when increasing (Left) model size, (Middle) sequence length (model size = 340M) (3) (Right) sequence length (model size = 760M) on C4 dataset."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.286,
+ 0.921,
+ 0.332
+ ],
+ "angle": 0,
+ "content": "sizes of 120M, 340M, 760M, and 1.3B parameters. We train small models (120M and 340M) on 15B tokens sampled from the dataset, the medium size model (760M) on 30B tokens, and the large model on 100B tokens. Baseline results are reported by Behrouz et al. (2024c)."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.111,
+ 0.35,
+ 0.617,
+ 0.368
+ ],
+ "angle": 0,
+ "content": "6.1 Language Modeling and Common-sense Reasoning"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.374,
+ 0.922,
+ 0.48
+ ],
+ "angle": 0,
+ "content": "We follow recent studies (Behrouz et al. 2024c; Yang et al. 2024a,c) and first focus on the perplexity in language modeling and also commonsense reasoning tasks. The results for MEMORA, YAAD, MONETA and also baselines with size of 340M, 760, and 1.3B are reported in Table 2. All of our variants outperforms all the baselines including Transformer++, modern linear recurrent models and hybrid methods. The superior performance compared to hybrid models is particularly important as all of our variants are pure recurrent (attention-free). Among the three variants of MirAS, while MONETA achieves slightly weaker performance than MEMORA, and YAAD, the other two variants are close and depending on the task and model size, the best model can vary."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.113,
+ 0.498,
+ 0.296,
+ 0.515
+ ],
+ "angle": 0,
+ "content": "6.2 Scaling Pattern"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.523,
+ 0.919,
+ 0.555
+ ],
+ "angle": 0,
+ "content": "To evaluate the scaling pattern of models and for comparing them with baseline, in this section, we plot their performance with varying the model size and the context window."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.567,
+ 0.92,
+ 0.675
+ ],
+ "angle": 0,
+ "content": "Context Length. We first vary the training context length from 2K to 32K for two versions of our model with size 340M and 760M. The results are reported in Figure 3 (Middle and Right). All three variants of Miras scales better than state-of-the-art baselines when increasing the context length. We attribute this superior performance to: (1) expressive memory architecture. Contrary to baselines like Mamba2 and GSA that uses vector- and matrix-valued memory, our variants are using 2-layer MLPs with more expressive power to learn from longer sequences. (2) The choice of retention gate and attentional bias: All of our three variants go beyond the standard attentional biases and retention gates. These choices can help the memory to better manage its fixed-size capacity."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.686,
+ 0.92,
+ 0.735
+ ],
+ "angle": 0,
+ "content": "Model Size. We also report the #FLOPs vs. perplexity of our models and baselines in Figure 3 (Left). All three variants outperforms all baselines given almost the same budget of FLOPs. These results, once again support the importance of powerful memory design."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.113,
+ 0.75,
+ 0.336,
+ 0.768
+ ],
+ "angle": 0,
+ "content": "6.3 Needle In Haystack"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.775,
+ 0.922,
+ 0.88
+ ],
+ "angle": 0,
+ "content": "To evaluate the effective context window of our models and baselines, we use needle-in-haystack task. In this task, we evaluate the model on retrieving a piece of information (i.e., the \"needle\") from long distractor texts (i.e., the \"haystack\"). We focus on the Single NIAH (S-NIAH) task from RULER benchmark (Hsieh et al. 2024) and evaluate our models and baselines on sequences with length 1K, 2K, 4K, and 8K. The results are reported in Table 3. All our variants outperforms all the baselines with a considerable margin. Interestingly, MONETA shows better performance than others when the data is synthetic noise (S-NIAH-PK). This observation validates the effectiveness of \\( p \\)-norm objective and retention gates as they are more robust to noise."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.509,
+ 0.938,
+ 0.526,
+ 0.949
+ ],
+ "angle": 0,
+ "content": "16"
+ }
+ ],
+ [
+ {
+ "type": "table_caption",
+ "bbox": [
+ 0.111,
+ 0.091,
+ 0.922,
+ 0.144
+ ],
+ "angle": 0,
+ "content": "Table 2: Performance of MIRAS's variants and baselines on language modeling and common-sense reasoning tasks. Hybrid models are marked with *. The best results of simple and hybrid models are highlighted. In largest scale, we compare our simple models with even hybrid models and highlight the best results."
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 0.158,
+ 0.156,
+ 0.877,
+ 0.761
+ ],
+ "angle": 0,
+ "content": "| Model | Memory Architecture | Attentional Bias | Retention Gate† | Memory Algorithm | Memory Write Operation |
| Shallow Memory |
| RetNet (2023) | Vector | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| Transformer (2017) | Matrix | L2 | - | Nonparametric | Mt=Mt-1∪{kt, vt} |
| LA (2021) | Matrix | Dot-Product | - | GD | Mt=Mt-1+vtktT |
| DFW | Matrix | Dot-Product | L2 | GD | Mt=(βtαT) ⊙ Mt-1+vtktT |
| Lightening Attention (2025) | Matrix | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| GLA (2024) | Matrix | Dot-Product | L2 | GD | Mt=Diag(αt)Mt-1+vtktT |
| Mamba (2024) | Matrix | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| HGRN2 (2024) | Matrix | L1 | L2 | GD | Mt=Diag(αt)Mt-1+vt(1-αt)T |
| DeltaNet (2017) | Matrix | L2 | - | GD | Mt=(I-βtktkT)Mt-1+βtvtktT |
| Longhorn (2024) | Matrix | L2 | - | Implicit GD | Mt=(I-βtktkT)Mt-1+(βt1+ktkβt)xtkT |
| TTT-Linear (2024) | Matrix | L2 | - | GD | Mt=Mt-1-η∇L(Mt-1, xt) |
| Gated DeltaNet (2024) | Matrix | L2 | L2 | GD | Mt=(αt(I-βtktkT))Mt-1+βtvtktT |
| RWKV-7 (2025) | Matrix | L2 | L2 | GD | Mt=diag(αt)(I-βtktkT)Mt-1+βtvtktT |
| DeltaProduct (2025) | Matrix | L2 | L2 | MGD* | Mt=(αtΠi=1n(I-βt,ikt,i)T)Mt-1+Σj=1nΠi=j(I-βt,ivtj,kj,i) |
| Deep Memory |
| TTT-MLP (2024) | 2-layer MLP | L2 | - | GD | Mt=Mt-1-η∇L(Mt-1;kt, vt) |
| Titans-LMM (2024) | k-layer MLP | L2 | L2 | GD + Momentum | Mt=αMt-1-St, where St=ηSt-1-θt∇L(Mt-1;kt, vt) |
| MONETA (ours) | 2-layer MLP | Lp | Lq | GD | At=AtA1-ηt∇lp(Wt-1;kt, vt), Wt=At/||At||q-2 |
| YAAD (ours) | 2-layer MLP | Huber | L2 | GD | Wt=atWt-1-(ηt∇ε2(Wt-1;kt, vt) if ||M(kt)-vt|≤δt, ηtδt∇ε1(Wt-1;kt, vt) Otherwise. |
| MEMORA (ours) | 2-layer MLP | L2 | KL | GD | Wt=Softmax(αt log(Wt-1)-ηt∇ε2(Wt-1;kt, vt)) |
+
+* is using multiple rounds of GD per token.
+For the sake of clarity, we use L2 for all modified L2-like regularizations. However, in fact, only Titans and RWKV-7 are using L2 retention gate (see Section 4)
+
+Deep Memory Module: Titans and Test Time Training. To overcome the limited memory and to extend the effective context length of deep sequence models, more recent studies focus on a new generation of architectures with deep memory module (Behrouz et al. 2024c; Sun et al. 2024). These architectures are built on the meta-learning perspective, where the memory is an MLP architecture that is updated using gradient descent (with momentum) (Behrouz et al. 2024c; Sun et al. 2024). Sun et al. (2024) further provide a unifying perspective that how linear and softmax attention are respectively parametric and non-parametric solutions of (kernel) regression loss but consider other modern linear RNNs outside of this class of models. Recently, in a concurrent work to ours, Wang et al. (2025) show that with additional simplification of modern RNNs (e.g., RetNet (Sun et al. 2023), Mamba (Dao et al. 2024)) they approximately place in the same class of models that internally optimize regression loss. It, however, still remains unanswered that "What is the underlying design framework behind these sequence models that can accurately unify existing architectures?" Moreover, the role of forget gates and its alternative choices in modern sequence models is surprisingly less explored.
+
+# 3 Associative Memory, Attentional Bias, and Retention
+
+Associative memory, which is an inseparable component of learning in humans (Terry 2017), has been the inspiration for many artificial neural architectures in the literature (Behrouz et al. 2024c; Hopfield 1982; Neil et al. 2017). These studies, however, define instances of the concept of associative memory, limiting the architecture to a specific class of similarity metrics between entities (i.e., keys and values). That is, broadly speaking, associative memory is an operator that maps a set of keys $K$ to a set of values $V$ , and so to learn the underlying mapping patterns in data, it requires an objective that targets a type of memory and measures the quality of learned mappings:
+
+Definition 3.1 (Associative Memory and Attentional Bias). Given a set of keys $\mathcal{K} \subseteq \mathbb{R}^{d_k}$ and values $\mathcal{V} \subseteq \mathbb{R}^{d_o}$ , associative memory is an operator $\mathcal{M}: \mathcal{K} \to \mathcal{V}$ . Learning the mapping of associative memory is based on an objective $\mathcal{L}$ , called
+
+Attentional Bias, that determines the type of memory and its tendency to prioritize some events:
+
+$$
+\mathcal {M} ^ {*} = \arg \min _ {\mathcal {M}} \quad \mathcal {L} (\mathcal {M} (\mathcal {K}); \mathcal {V}). \tag {4}
+$$
+
+A few remarks are in order:
+
+Remark 1. When we parameterize the memory with parameter $W$ , we use $\mathcal{M}(W, \mathbf{k})$ . In this parametric setting, the optimization problem in (4) should be performed over the parameter $W$ . Furthermore, in the parametric setup, we might use an additional regularization $\mathcal{R}(W)$ to control the retaining of the past data.
+
+Remark 2. Learning the mapping between keys and values (Equation 4) is a meta-learning problem, in which the attentional bias is optimized in the inner-loop and all other parameters of the neural network (e.g., linear projections, convolutions, etc.) are optimized in the outer-loop. Therefore, the model learns how to store the data into its parameters at test time (Behrouz et al. 2024c; Sun et al. 2024).
+
+# 3.1 Learning to Memorize and to Retain Through the Lens of Optimization
+
+Definition 3.1 translates the design of a neural architecture based on the concept of associative memory to learning the underlying mapping between keys and values, by minimizing an objective $\mathcal{L}$ . To optimize Equation 4, one simple approach is to utilize the idea of gradient descent. Specifically, given a new pair of keys and values, we update the memory as:
+
+$$
+W _ {t} = W _ {t - 1} - \eta_ {t} \nabla \ell \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right), \tag {5}
+$$
+
+where, for simplicity, we use the definition $\ell(W_{t-1}; \mathbf{k}_t, \mathbf{v}_t) \coloneqq \mathcal{L}(\mathcal{M}(W; \mathbf{k}_t), \mathbf{v}_t)$ . Behrouz et al. (2024c) re-interpret the formulation as a momentary surprise metric, where the model memorizes tokens that violates the expectation of the objective (i.e., being surprising to the memory). Although the choice of objective is an important step to fully interpret Equation 5 (which we discuss in detail in Section 5), there are different viewpoints to interpret this update rule in its general format, which later can help us to go beyond existing architectures:
+
+# 3.2 Viewpoint 1: Online Regression and Follow-The-Regularized-Leader
+
+Equation (5) can be viewed as one step of online gradient descent over the sequence of the loss functions
+
+$$
+\ell \left(W; \mathbf {k} _ {1}, \mathbf {v} _ {1}\right), \ell \left(W; \mathbf {k} _ {2}, \mathbf {v} _ {2}\right), \dots , \ell \left(W; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right), \dots . \tag {6}
+$$
+
+It is well known that the online gradient descent can be viewed as a special case of Follow-The-Regularized-Leader (FTRL) algorithm with a special choice of loss functions (Shalev-Shwartz et al. 2012, Chapter 2) and (Hazan et al. 2016). Specifically, assuming $W_0 = 0$ , the update rule in (5) is equivalent to
+
+$$
+W _ {t} = \arg \min _ {W} \quad \sum_ {i = 1} ^ {t} \left\langle W - W _ {i - 1}, \nabla \ell \left(W _ {i - 1}; \mathbf {k} _ {i}, \mathbf {v} _ {i}\right) \right\rangle + \frac {1}{2 \eta} \| W \| _ {2} ^ {2}, \tag {7}
+$$
+
+where the term $\langle W - W_{i-1}, \nabla \ell(W_{i-1}; \mathbf{k}_i, \mathbf{v}_i) \rangle$ is the local linear approximation of the original loss at time $i$ and the second term is a regularization term. While the first part $\sum_{i=1}^{t} \langle W - W_{i-1}, \nabla \ell(W_{i-1}; \mathbf{k}_i, \mathbf{v}_i) \rangle$ measures how well can the memory learn all the past tokens, the second term $\frac{1}{2\eta} \|W\|_2^2$ penalizes the memory update with respect to the size of memory.
+
+Equation (7) uses linear approximation of the loss function and quadratic regularization. We can, however, in principle use other approximations of the loss function as well as other regularization functions, as used in the past in online optimization (Hazan et al. 2016; Shalev-Shwartz et al. 2012) or in general optimization (Miral 2015; Razaviyayn et al. 2013). Such changes are the idea behind the development of other optimization algorithms such mirror descent. More specifically, we can generalize the update rule in (7) to the form:
+
+$$
+W _ {t} = \arg \min _ {W \in \mathcal {W}} \underbrace {\sum_ {i = 1} ^ {t} \widehat {\ell_ {i}} (W ; \mathbf {k} _ {i} , \mathbf {v} _ {i})} _ {\text {A t t e n t i o n a l B i a s}} + \underbrace {\frac {1}{\eta_ {t}} \mathcal {R} _ {t} (W)} _ {\text {M e m o r y S t a b i l i t y}}. \tag {FTRLViewpoint}
+$$
+
+In this update rule, the term $\sum_{i=1}^{t} \widehat{\ell}_i(W; \mathbf{k}_i, \mathbf{v}_i)$ aims at memorizing the tokens at test time, while the term $\mathcal{R}_t(W)$ regularizes the learning dynamics and take the size of the memory into account when updating it by a new incoming data. Choosing different loss functions $\widehat{\ell}_i(W; x_i)$ and the regularization term $\frac{1}{\eta_t} \mathcal{R}_t(W)$ can lead to different algorithms such as (online) gradient descent or mirror descent. In this generalization, $\eta_t$ to can be data-dependent. Moreover, we will allow imposing constraint $\mathcal{W}$ on the choice $W$ .
+
+# 3.3 Viewpoint 2: Learning the Latest Token While Retaining Previous Information
+
+Another way to interpret the update rule (5) is to view it as learning from the latest key-value pair $(\mathbf{k}_i, \mathbf{v}_i)$ (via using its gradient or surprise metric), while staying close to the previous state $W_{t-1}$ to retain the previously memorized tokens. Formally, (5) is equivalent to
+
+$$
+W _ {t} = \arg \min _ {W} \left\langle W - W _ {t - 1}, \nabla \ell (W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}) \right\rangle + \frac {1}{2 \eta_ {t}} \left\| W - W _ {t - 1} \right\| _ {2} ^ {2}
+$$
+
+The first term locally approximates $\ell(W; \mathbf{k}_t, \mathbf{v}_t)$ around the previous state $W_{t-1}$ , while the last term regularizes deviations from $W_{t-1}$ . This form can generalize to
+
+$$
+W _ {t} = \arg \min _ {W \in \mathcal {W}} \underbrace {\widetilde {\ell_ {t}} (W ; \mathbf {k} _ {t} , \mathbf {v} _ {t})} _ {\text {A t t e n t i o n a l B i a s}} + \underbrace {\operatorname {R e t} _ {t} (W , W _ {t - 1})} _ {\text {R e t e n t i o n}}, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text {(L e a r n i n g - R e t a i n i n g V i e w p o i n t)}
+$$
+
+where the term $\widetilde{\ell_t} (W;\mathbf{k}_t,\mathbf{v}_t)$ is an approximation of $\ell (W;\mathbf{k}_t,\mathbf{v}_t)$ and minimizing it corresponds to Learning from the new concepts $(\mathbf{k}_t,\mathbf{v}_t)$ . The second term $\mathrm{Ret}_t(W,W_{t - 1})$ regularizes the changes in $W$ to make the learning dynamics stable and to retain previously learned knowledge. This Retention function may have local and global components:
+
+$$
+\operatorname {R e t} _ {t} \left(W, W _ {t - 1}\right) = \underbrace {\frac {1}{\eta_ {t}} \mathrm {D} _ {t} \left(W , W _ {t - 1}\right)} _ {\text {L o c a l R e t e n t i o n}} + \underbrace {\frac {1}{\alpha_ {t}} \mathrm {G} _ {t} \left(W\right)} _ {\text {G l o b a l R e t e n t i o n}}.
+$$
+
+Here, the term $\mathrm{D}_t(W, W_{t-1})$ , which is a premetric that controls the deviations from $W_{t-1}$ , aims at retaining previously learned knowledge. The coefficient $\eta_t$ can be viewed as a meta in-context learning rate, where larger values of $\eta_t$ leads to learning more from new concepts, while allowing higher forgetting of previously learned concepts. The second term is a global retention that controls the change of the memory with respect to its size. The special instances of the above viewpoint (e.g., without global retention, with implicit closed-form solution, and/or with limited memory structure) have been the motivation behind some of the recent studies such as Liu et al. (2024a).
+
+# 3.4 Further Discussions on the Two Viewpoints
+
+The (FTRL Viewpoint) and (Learning-Retaining Viewpoint) are connected through the lens of online optimization. For example, as discussed above, by choosing linear approximation of the loss and quadratic regularization/retention, they can both cover online gradient descent update in (5) as a special case. One straightforward way to make the connection explicit is by defining the premetric $\mathrm{D}_t(W;W^{\prime})$ based on the previous loss functions and the regularization, as described in Proposition 3.2 below:
+
+Proposition 3.2. Let $\eta_t = \eta$ and define $h_t(W) \coloneqq \sum_{i=1}^{t-1} \widehat{\ell}_i(W; \mathbf{k}_i, \mathbf{v}_i) + \frac{1}{\eta} R(W)$ . Assume $\mathcal{W} = \mathbb{R}^d$ and the function $h_t(W)$ is strictly convex in $W$ and let $\mathcal{D}_h(\cdot, \cdot)$ be the Bregman divergence defined by function $h(\cdot)$ , i.e., $\mathcal{D}_h(W, W') = h(W) - h(W') - \langle \nabla h(W'), W - W' \rangle$ . Set $Ret_t(W, W') = \mathcal{D}_h(W, W')$ and $\widetilde{\ell}_t(W; x_t) = \widehat{\ell}_t(W; x_t)$ in (Learning-Retaining Viewpoint). Then, the update rule in (Learning-Retaining Viewpoint) is equivalent to the update rule in (FTRL Viewpoint).
+
+We provide the proof in Appendix B. The above proposition shows that (Learning-Retaining Viewpoint) can also explain the approaches obtained by (FTRL Viewpoint), under some mild assumptions. Hence, (Learning-Retaining Viewpoint) may be seen as a more general version. This is why we focus on this viewpoint in most of our derivations in the next sections.
+
+Remark 3. Given the above viewpoint, we can see that even by using additional global regularization there is no memory erasing or forgetting process (a common term in modern architectures (Behrouz et al. 2024c; Yang et al. 2024a)) but the model might decide to not retain the past state of the memory. Interestingly, this observation also matches the human memory process, where brain does not erase memories but they might become inaccessible due to retrieval failures (Robertson 2002). Therefore, instead of calling it a forget gate, later on, we use "Retention Gate" to refer to this term.
+
+Remark 4. As we discuss in Section 4 and summarize in Table 1, most existing modern sequence models are optimizing associative memory objective (attentional bias in Equation 4) using gradient descent. Therefore, to provide further intuition about the connection of existing sequence models as well as their online learning interpretations, we discuss the above two viewpoints that are limited to gradient descent-based update rules. Our initial definition of attentional bias and associative memory in Equation 4, however, is broader and can be optimized by any optimization algorithm (e.g., even Newton's method, or non-parametric solutions).
+
+# 4 MirAs: Learning to Memorize with Robust and Expressive Memory
+
+Building upon our definition of associative memory, attentional bias, and previous viewpoints, we present MIRAs framework that not only accurately unifies existing backbone architectures but it also provides insights on how to design the next generation of sequence models. As discussed earlier in Section 3, learning an associative memory can be interpreted as a meta-learning task, in which the associative memory learns how to compress and store data into its parameters at test time. The architecture of the memory in such tasks is particularly important as in longer contexts, the expressivity of the memory structure can limit its ability to learn the underlying patterns. Therefore, the first choice to design a sequence model is the structure of the memory. Given the structure of the memory, parameterized by a set of parameters $W$ , as discussed earlier, we aim to minimize a loss function $\ell(W; \cdot, \cdot)$ with a retention regularizer $\mathrm{Ret}(\cdot)$ via a learning algorithm (e.g., gradient descent). Accordingly, MIRAs requires four design choices:
+
+1. Memory Structure: This choice specifies the architecture of the memory. For example, this architecture can be a vector, a linear function, a Multilayer Perceptron (MLP) layer, or even more complex structures. We may restrict the choice of $W$ to be within a certain region, e.g., $W$ to lie within an $L_{2}$ ball to avoid infinite values or unstable training.
+2. Attentional Bias: A key choice is the attentional bias objective $\mathcal{L}(\cdot)$ in Equation 4. We can even consider different approximations of the loss function, (e.g., $\widehat{\ell} (\cdot ,\cdot)$ in (FTRL Viewpoint) or $\widetilde{\ell} (\cdot ,\cdot)$ in (Learning-Retaining Viewpoint)). The choice of attentional bias determines how memory memorizes the context, maps the inputs, and prioritizes the events.
+3. Memory Stability and Retention: Another key choice is the retention regularizer $\mathcal{R}(\cdot)$ (e.g., $\mathcal{R}_t(\cdot)$ in (FTRL Viewpoint) and $\mathrm{Ret}_t(\cdot)$ in (Learning-Retaining Viewpoint)). In parametric setups, this choice balances learning with retention of past state. An effective retention gate is key to the good performance in long context tasks.
+4. Memory Algorithm: Finally, this choice specifies the learning algorithm that we use to optimize the memory objective. One may use gradient descent, gradient descent with momentum, or any other algorithm (including finding non-parametric solutions).
+
+The above choices are major design choices for designing backbone sequence models in neural architectures. There are, however, minor decisions that can distinguish models; i.e., data-dependent or independent parameters, scalar or channel-wise learning rate/retaining gate, etc. Next, we discuss the overview of how existing architectures fit into MIRAS framework.
+
+RNNs with Hebbian Rule. The first generation of modern recurrent architectures (e.g., Linear attention (Katharopoulos et al. 2020), RetNet (Sun et al. 2023), Mamba (Gu et al. 2024), and GLA (Yang et al. 2024b)) are based on Hebbian-like (e.g., gated Hebbian) learning rule (Hebb 2005). We let attentional bias be the dot product similarity. That is, given a memory $\mathcal{M} \in \mathbb{R}^{d \times n}$ and $\mathbf{k}, \mathbf{v} \in \mathbb{R}^d$ , we define $\tilde{\ell}_t \coloneqq -2\langle \mathcal{M}_t \mathbf{k}_t, \mathbf{v}_t \rangle$ and local retention as $\mathrm{Ret}_t(\mathcal{M}, \mathcal{M}_{t-1}) = \| \mathcal{M}_t - \alpha \mathcal{M}_{t-1} \|_F^2$ . Using Equation Learning-Retaining Viewpoint and gradient descent as the optimizer (i.e., memory learning algorithm), the memory update rule is:
+
+$$
+\mathcal {M} _ {t} = \alpha \mathcal {M} _ {t - 1} + \mathbf {v} _ {t} \mathbf {k} _ {t} ^ {\top}. \tag {8}
+$$
+
+When (1) $\alpha = 1$ , memory update is equivalent to Linear Attention (LA) (Katharopoulos et al. 2020); (2) $\alpha \in \mathbb{R}$ is a learnable parameter, resulting architecture is either lightening attention ( $n > 1$ ) (Li et al. 2025) or RetNet ( $n = 1$ ) (Sun et al. 2023); and (3) $\alpha_{t} \in \mathbb{R}$ are data-dependent learnable parameters, resulting sequence model is Mamba2 (Dao et al. 2024).
+
+RNNs with Delta Rule. To improve the memory management and to enhance the memory capacity of the above group, several studies suggest using delta rule (Neil et al. 2017; Schlag et al. 2021) as the learning algorithm in recurrent neural networks (e.g., DeltaNet (Schlag et al. 2021), Longhorn (Liu et al. 2024a), and RWKV7 (Peng et al. 2025b)). In this part, we recall that where $\mathcal{M} \in \mathbb{R}^{d \times n}$ , delta rule is equivalent to optimizing MSE objective $\| \mathcal{M}_t \mathbf{k}_t - \mathbf{v}_t \|_2^2$ with $\mathrm{Ret}_t(\mathcal{M}, \mathcal{M}_{t-1}) = \| \mathcal{M}_t - \alpha \mathcal{M}_{t-1} \|_F^2$ as local retention, and stochastic gradient descent as optimizer: ( $\eta_t$ is defined in Equation Learning-Retaining Viewpoint)
+
+$$
+\mathcal {M} _ {t} = \alpha \left(\mathbf {I} - \eta_ {t} \mathbf {k} _ {t} \mathbf {k} _ {t} ^ {\top}\right) \mathcal {M} _ {t - 1} + \mathbf {v} _ {t} \mathbf {k} _ {t} ^ {\top}. \tag {9}
+$$
+
+When (1) $\alpha = 1$ , memory update is equivalent to DeltaNet (Schlag et al. 2021); and (2) $\alpha_{t} \in \mathbb{R}^{m}$ are data-dependent learnable parameters, resulting sequence model is either Gated DeltaNet (Yang et al. 2024a) ( $m = 1$ ), or RWKV7 (Peng et al. 2025b) ( $m = d$ ). Therefore, RNNs with delta rule are special instances of MIRAS.
+
+Beyond Delta Rule. As discussed earlier, while delta rule with its value replacement strategy is more powerful than Hebbian-like learning rules, it suffers from theoretical limitations (Irie et al. 2023) and achieves moderate performance in practice (Yang et al. 2024c). Therefore, several studies have focused on update rules beyond delta rule. Recently, Titans (Behrouz et al. 2024c) suggests using non-linear MSE objective of $\| \mathcal{M}_t(\mathbf{k}_t) - \mathbf{v}_t\| _2^2$ with both local and global retention of $\mathrm{D}_t = \| W_t - W_{t - 1}\| _F^2$ and $\mathrm{G}_t = \| W_t\| _2^2$ and optimize it with gradient descent with momentum $^2$ . Therefore, Titans-LMM is a special instance of MIRAs, where we use the abovementioned attentional bias and retention regularizations, and gradient descent with momentum as the optimizer.
+
+Another example of such models is Mesa-layer, in which the model uses $\sum_{i=1}^{t} \|\mathcal{M}_{t}(\mathbf{k}_{i}) - \mathbf{v}_{i}\|_{2}^{2}$ as the attentional bias objective with $\|\mathcal{M}_{t}\|_{2}^{2}$ as the retention regularization. Since these models use Newton's method to optimize such an objective, they provide a more expressive update rule than delta rule. We further discuss a set of new learning algorithms beyond delta rule in Section 5.
+
+Attention. As discussed by Sun et al. (2024), softmax attention is a non-parametric solution of $\ell_2$ -MSE loss function (i.e., $\| W\mathbf{k} - \mathbf{v}\| _2^2$ ) with Nadaraya-Watson estimator. Therefore, softmax attention is an instance of MIRAS, when we find the non-parametric solution to the MSE loss with Nadaraya-Watson estimator, without retention.
+
+# 5 Beyond Existing Attentional Biases and Retention Gates
+
+As discussed in the previous section, existing work focuses only on linear/quadratic choices for the attentional bias or retention gate. In particular, the loss function $L(\mathcal{M}(\mathbf{k}_t),\mathbf{v}_t)$ is defined as $L(\mathcal{M}(\mathbf{k}_t),\mathbf{v}_t) = c_t\| \mathcal{M}(\mathbf{k}_t) - \mathbf{v}_t\|^2$ for some (learnable) constant $c_{t}$ in prior work. Also the regularization term $R_{t}(W)$ or the parametric $D_{t}$ is considered as a quadratic/linear function. In addition, almost all prior work considers $W$ to be the entire $\mathbb{R}^d$ space. However, in general there could be various choices for all the three aforementioned design choices. To illustrate the potential and flexibility of our designed framework, here, we review some of the potential design choices for attentional bias and retention gate in MirAS. For the sake of clarity, we discuss all these attentional bias and memory retention gates based on using gradient descent as the optimizer, and so based on the provided two view points. However, these attentional bias objectives and retention regularizers can be directly used in Equation 4 and optimized by using any other optimization algorithms, resulting in different update rules.
+
+# 5.1 Alternative Attentional Biases
+
+Variant 1: $\ell_p$ -Attentional Bias. As discussed in the main body, attentional bias defines the "similarity metric" and measures how well memory can recall the value, given its corresponding key. Although $\ell_2$ regression loss often is a natural choice, it is sensitive to noise in the data. A natural extension is to use $\ell_p$ -norm class of objectives. That is, let $\mathcal{M}$ be the memory, $\mathbf{k}$ be the keys, and $\mathbf{v}$ be the values, we define $\ell_p$ -attentional bias as:
+
+$$
+\mathcal {L} \left(\mathcal {M} \left(W, \mathbf {k} _ {t}\right); \mathbf {v} _ {t}\right) = \| \mathcal {M} \left(\mathbf {k} _ {t}\right) - \mathbf {v} _ {t} \| _ {p} ^ {p}, \tag {10}
+$$
+
+where $p \in \mathbb{R}^{\geq 1}$ and $\| . \|_p$ is the $p$ -norm. Although depending on the distribution of the data, we might want to use different values of $p$ (see Section 6), different values of $p$ can result in memory architectures with interesting properties. For the sake of simplicity, let memory be a matrix, i.e., $W \in \mathbb{R}^{m \times d}$ and $\mathcal{M}(W, \mathbf{k}_t) = W\mathbf{k}_t$ , the closed form can be derived as:
+
+$$
+W _ {t} = W _ {t} - \eta_ {t} \nabla \ell \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right) = W _ {t} - p \eta_ {t} \left(\operatorname {S i g n} \left(W \mathbf {k} _ {t} - \mathbf {v} _ {t}\right) \odot | W \mathbf {k} _ {t} - \mathbf {v} _ {t} | ^ {p - 1}\right) \mathbf {k} _ {t} ^ {\top}. \tag {11}
+$$
+
+Let $p = 1$ , the recurrence is simplified as:
+
+$$
+W _ {t} = W _ {t} - \eta_ {t} \operatorname {S i g n} \left(W _ {t} \mathbf {k} _ {t} - \mathbf {v} _ {t}\right) \mathbf {k} _ {t} ^ {\top}, \tag {12}
+$$
+
+which means that the memory has only two values of $-1$ and $1$ . We call this variation value-less associative memory, in which we store entities (keys) but map them into two extreme class of $-1$ and $+1$ .
+
+Remark 5. One of the critical challenges to use the above update rule is in the backpropagation process, in which $\operatorname{Sign}(\cdot)$ and $|\cdot|$ are non-differentiable and so might cause unstable training. To overcome this issue, we use $\operatorname{Sign}(x) \approx \tanh(\alpha x)$ , and $|x| = \sqrt{x^2 + \epsilon}$ , as the smooth approximators of these functions.
+
+One simple interpretation for such behavior (i.e., value-less memory) is similar to the coping mechanism in humans (Loftus 1993), in which the memory does not store the values for extreme events. This interpretation of protective memory in extreme events motivates our next variant.
+
+Variant 2: Huber Loss: Memory with Coping Mechanism. While $\ell_2$ -norm objective is a common choice for many statistical and machine learning tasks, it is known to be sensitive to outliers and extreme samples. This sensitivity extends to the use of $\ell_2$ loss for attentional bias. To address this and drawing motivation from robust regression literature, we suggest utilizing the Huber loss-type (Hastie et al. 2009; Huber 1992) as the attentional bias, thereby reducing the negative impact of the outlier data on the memory learning process.
+
+We can apply Huber-type loss in three different ways: The first approach is to define the summation of the Huber loss across different coordinates as the total loss, i.e.,
+
+$$
+\ell (W; \mathbf {k} _ {t}, \mathbf {v} _ {t}) = \sum_ {j} \mathcal {H} (\mathcal {M} (W, \mathbf {k} _ {t}) _ {j} - \mathbf {v} _ {t, j}),
+$$
+
+where $\mathcal{M}(W,\mathbf{k}_t)_j$ and $\mathbf{v}_{t,j}$ denote the $j$ -th coordinate of $\mathcal{M}(W,\mathbf{k}_t)$ and $\mathbf{v}_t$ respectively. The function $\mathcal{H}(\cdot):\mathbb{R}\mapsto \mathbb{R}$ is the Huber loss defined as
+
+$$
+\mathcal {H} (a) = \left\{ \begin{array}{l l} \frac {1}{2} a ^ {2} & \text {i f} | a | \leq \delta \\ \delta \left(| a | - \frac {1}{2} \delta\right) & \text {i f} | a | > \delta . \end{array} \right. \tag {13}
+$$
+
+Utilizing this attentional bias can lead to various memory update rules. For example, for the matrix form memory $\mathcal{M}(W,\mathbf{k}_t) = W\mathbf{k}_t$ , the update rule is given by
+
+$$
+W _ {t} = W _ {t - 1} - \eta_ {t} \left[ \left(\left(W \mathbf {k} _ {t} - \mathbf {v} _ {t}\right) \mathbf {k} _ {t} ^ {T}\right) \odot \left(\mathbf {I} \left(\left| W \mathbf {k} _ {t} - \mathbf {v} _ {t} \right| \leq \delta_ {t}\right) \mathbf {1} ^ {\top}\right) + \left(\delta_ {t} \operatorname {S i g n} \left(W \mathbf {k} _ {t} - \mathbf {v} _ {t}\right) \mathbf {k} ^ {\top}\right) \odot \left(\mathbf {I} \left(\left| W \mathbf {k} _ {t} - \mathbf {v} _ {t} \right| > \delta_ {t}\right) \mathbf {1} ^ {\top}\right) \right] \tag {14}
+$$
+
+In this formulation, the parameter $\delta_t$ decides the type of the memory used for each block of memory ( $\ell_2$ -norm objective or value-less) based on the context, making the memory more robust to outliers.
+
+The second approach is to define the Huber-type loss based on the $\ell_2$ loss over all coordinates, i.e.,
+
+$$
+\ell (W; \mathbf {k} _ {t}, \mathbf {v} _ {t}) = \mathcal {H} (\| \mathcal {M} (W, \mathbf {k} _ {t}) - \mathbf {v} _ {t} \| _ {2}).
+$$
+
+For simplicity of derivations, assume matrix memory $M(W,\mathbf{k}_t) = W\mathbf{k}_t$ . Then using gradient descent for updating memory leads the memory update rule
+
+$$
+W _ {t} = W _ {t - 1} - \eta_ {t} \left\{ \begin{array}{l l} \left(\mathcal {M} \left(W _ {t - 1}, \mathbf {k} _ {t}\right) - \mathbf {v} _ {t}\right) \mathbf {k} _ {t} ^ {T} & \text {i f} \| \mathcal {M} \left(W _ {t - 1}, \mathbf {k} _ {t}\right) - \mathbf {v} _ {t} \| _ {2} \leq \delta_ {t}, \\ \delta_ {t} \frac {\left(\mathcal {M} \left(W _ {t - 1} , \mathbf {k} _ {t}\right) - \mathbf {v} _ {t}\right)}{\| \mathcal {M} \left(W _ {t - 1} , \mathbf {k} _ {t}\right) - \mathbf {v} _ {t} \| _ {2}} \mathbf {k} _ {t} ^ {T} & \text {O t h e r w i s e .} \end{array} \right. \tag {15}
+$$
+
+Again, in the form (15), the parameter $\delta_t$ decides the type of the memory used ( $\ell_2$ -norm objective or normalized version) based on the context, making the memory more robust to outliers.
+
+Finally, in the third approach, we present a smooth mixture method, in which the memory decides if for an incoming data it is better to use $\ell_2$ or $\ell_1$ attentional bias:
+
+$$
+W _ {t} = W _ {t - 1} - \left\{ \begin{array}{l l} \eta_ {t} \nabla \ell_ {2} \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right) & \text {i f} \| \mathcal {M} (\mathbf {k} _ {t}) - \mathbf {v} _ {t} \| \leq \delta_ {t}, \\ \eta_ {t} \delta_ {t} \nabla \ell_ {1} \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right) & \text {O t h e r w i s e .} \end{array} \right. \tag {16}
+$$
+
+The role of parameter $\delta_t$ is the same as above.
+
+Variant 3: Memory Robust to Value Shifts. Following the robustness requirement discussed in the previous section, we aim to design a memory mechanism that exhibits resilience against small shifts in the value parameter. A natural approach in this context is to employ a robust optimization formulation. Specifically, we define the loss function as the worst-case $\ell_2$ distance between the predicted memory output and the perturbed true value:
+
+$$
+\mathcal {L} \left(\mathcal {M} \left(W, \mathbf {k} _ {t}\right); \mathbf {v} _ {t}\right) = \max _ {\| \delta \mathbf {v} _ {t} \| _ {2} \leq \Delta} \frac {1}{2} \| \mathcal {M} \left(W, \mathbf {k} _ {t}\right) - \left(\mathbf {v} _ {t} + \boldsymbol {\delta} \mathbf {v} _ {t}\right) \| _ {2} ^ {2}. \tag {17}
+$$
+
+This formulation seeks the memory parameters $W$ that perform well even under the adverse local perturbation of the true value $\mathbf{v}_t$ within an $\ell_2$ ball of radius $\Delta$ . To solve the maximization problem in (17), we find the optimal perturbation $\delta \mathbf{v}_t^*$ . By solving this problem with respect to $\delta \mathbf{v}_t$ , we arrive at:
+
+$$
+\delta \mathbf {v} _ {t} ^ {*} = \Delta \frac {- \mathcal {M} (W , \mathbf {k} _ {t}) + \mathbf {v} _ {t}}{\| \mathcal {M} (W , \mathbf {k} _ {t}) - \mathbf {v} _ {t} \| _ {2}}
+$$
+
+Substituting this optimal perturbation back into the loss function (17), we obtain the robust loss:
+
+$$
+\mathcal {L} \left(\mathcal {M} \left(W, \mathbf {k} _ {t}\right); \mathbf {v} _ {t}\right) = \frac {1}{2} \| \mathcal {M} \left(W, \mathbf {k} _ {t}\right) - \mathbf {v} _ {t} \| _ {2} ^ {2} + \Delta \| \mathcal {M} \left(W, \mathbf {k} _ {t}\right) - \mathbf {v} _ {t} \| _ {2} + \frac {1}{2} \Delta^ {2}.
+$$
+
+This robust loss function is a combination of the standard $\ell_2$ loss and a term proportional to the $\ell_2$ norm of the error, scaled by the robustness parameter $\Delta$ . The value of $\Delta$ thus controls the trade-off between fitting the nominal data and ensuring robustness against value perturbations.
+
+For simplicity of the derivations, let us consider a constant value for $\Delta$ , an Euclidean retention gate $\mathrm{Ret}_t(W,W_{t - 1}) = \| W - W_{t - 1}\|^2$ , and an attentional bias term $\widetilde{\ell} (W;\mathbf{k}_t,\mathbf{v}_t) = \langle W - W_{t - 1},\nabla \ell (W_{t - 1};\mathbf{k}_t,\mathbf{v}_t)\rangle$ . Furthermore, to simplify the memory operation, we assume a linear matrix memory model $\mathcal{M}(W,\mathbf{k}_t) = W\mathbf{k}_t$ . Under these assumptions, we can derive the memory update mechanism using gradient descent on the robust loss:
+
+$$
+W _ {t} = W _ {t - 1} - \eta \left(\left(\mathcal {M} \left(W _ {t - 1}, \mathbf {k} _ {t}\right) - \mathbf {v} _ {t}\right) \mathbf {k} _ {t} ^ {\top} + \Delta \frac {\mathcal {M} \left(W _ {t - 1} , \mathbf {k} _ {t}\right) - \mathbf {v} _ {t}}{\| \mathcal {M} \left(W _ {t - 1} , \mathbf {k} _ {t}\right) - \mathbf {v} _ {t} \| _ {2}} \mathbf {k} _ {t} ^ {\top}\right)
+$$
+
+In this update rule, the parameter $\Delta$ , which governs the influence of the robustness term, can also be treated as a learnable parameter, allowing the model to adapt its robustness based on the observed data.
+
+# 5.2 Alternative Retention Gates
+
+Variant 1: Memorization Over A Scaled Probability Simplex Via $f$ -Divergence. A common technique in learning to prevent numerical instabilities and exploding values is to restrict the search space to a bounded domain. Following this principle, to avoid numerical instabilities, we can constrained the variable $W_{t}$ to lie within a (scaled) probability simplex. In other words, we can restrict the state to lie in the constraint set
+
+$$
+\mathcal {W} = \{W \mid \| W \| _ {1} = c \text {a n d} W _ {j l} \geq 0, \forall j, l \}.
+$$
+
+In this set, each matrix $W$ can be viewed as a measure. Thus, in (Learning-Retaining Viewpoint), we can utilize divergences over measures to define our premetric. For example, we can use $f$ -divergence measure (Polyanskiy et al. 2025, Def 4.9), (Csiszar 1967) to define $\mathrm{D}_t(\cdot, \cdot)$ . More specifically, let $f(\cdot)$ be a smooth strictly convex function from $\mathbb{R}^+$ to $\mathbb{R}$ with $f(1) = 0$ . Then, we can define the $f$ -divergence between $W$ and $W'$ as
+
+$$
+\mathrm {D} _ {t} (W, W ^ {\prime}) = \sum_ {j l} W _ {j l} ^ {\prime} f \left(\frac {W _ {j l}}{W _ {j l} ^ {\prime}}\right).
+$$
+
+It is known that $f$ -divergence is zero if and only if $W = W'$ ; see Polyanskiy et al. 2025, Theorem 2.3. Using the above premetric as the retention gate and setting $\widetilde{\ell}(W; \mathbf{k}_t, \mathbf{v}_t) = \langle W - W_{t-1}, \nabla \ell(W_{t-1}; \mathbf{k}_t, \mathbf{v}_t) \rangle$ in (Learning-Retaining Viewpoint), we get the update rule
+
+$$
+W _ {t} = W _ {t - 1} \odot g \left(- \zeta_ {t} - \eta_ {t} \nabla \ell \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right)\right). \tag {18}
+$$
+
+Here $g(\cdot)$ is the inverse of the mapping $f'$ , i.e., $g(f'(\tau)) = \tau$ , $\forall \tau$ ; the operator $\odot$ denotes the Hadamard (elementwise) product, and $\zeta_t$ should be chosen such that $\| W_t\|_1 = c$ . Notice that since the function $f(\cdot)$ is strictly convex and smooth, its derivative is strictly increasing and hence $g(\cdot)$ is well defined. Conversely, for any strictly monotone function $g(\cdot)$ , we can find its inverse function $g^{-1}$ (which is strictly increasing) and define $f(\tau) = \mathrm{const} + \int_{\tau' = 0}^{\infty}g^{-1}(\tau')d\tau'$ . The term const should be chosen such that $f(1) = 0$ . Then the update rule in (18) can be interpreted by the $f$ -divergence regularization, as explained above. Therefore, one can directly choose a continuous monotonically increasing function $g(\cdot)$ and use (18) for memory update.
+
+Specializing to KL divergence. Let us further make the above update rule explicit by using special function $f$ . If we choose $f(\tau) = \tau \ln(\tau)$ , then the $f$ -divergence becomes the widely used KL divergence measure $D_t(W, W_{t-1}) = \sum_{jl} W_{jl} \log \left( \frac{W_{jl}}{(W_t)_{jl}} \right)$ . In addition, we can also utilize the Shannon entropy as the global retention by regularizing deviations from uniform distribution, i.e., $G_t(W) = \sum_{jl} W_{jl} \log (W_{jl})$ . Combining these choices of the local and global retention gates, we obtain the overall retention gate
+
+$$
+\operatorname {R e t} _ {t} (W, W _ {t - 1}) = \frac {1}{\eta_ {t}} \sum_ {j l} W _ {j l} \log \left(\frac {W _ {j l}}{\left(W _ {t}\right) _ {j l}}\right) + \frac {1}{\alpha_ {t}} \sum_ {j l} W _ {j l} \log \left(W _ {j l}\right)
+$$
+
+Choosing the attentional bias $\widetilde{\ell}(W; \mathbf{k}_t, \mathbf{v}_t) = \langle W - W_{t-1}, \nabla \ell(W_{t-1}; \mathbf{k}_t, \mathbf{v}_t) \rangle$ and the above retention gate will lead to the update rule
+
+$$
+W _ {t} = \arg \min _ {W} \left\langle W - W _ {t - 1}, \nabla \ell \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right) \right\rangle + \frac {1}{\eta_ {t}} \sum_ {j l} W _ {j l} \log \left(\frac {W _ {j l}}{\left(W _ {t}\right) _ {j l}}\right) + \frac {1}{\alpha_ {t}} \sum_ {j l} W _ {j l} \log \left(W _ {j l}\right) \tag {19}
+$$
+
+$$
+\text {s . t .} \quad \sum_ {j l} W _ {j l} = c, W _ {j l} \geq 0, \forall j l \tag {20}
+$$
+
+Attaching the Lagrange multiplier to the first constraint, the KKT conditions imply
+
+$$
+\left(\nabla \ell \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right)\right) _ {j l} + \left(\frac {1}{\eta_ {t}} + \frac {1}{\alpha_ {t}}\right) \left(1 + \log W _ {j l}\right) - \frac {1}{\eta_ {t}} \log \left(\left(W _ {t - 1}\right) _ {j l}\right) + \mu_ {t} = 0, \quad \forall j, l
+$$
+
+where $\mu_t$ should be chosen such that $\sum_{jl} W_{jl} = c$ . Rearranging the terms and defining $\lambda_t = \frac{1 / \alpha_t}{1 / \alpha_t + 1 / \eta_t}$ , $\eta_t' = \frac{1}{1 / \alpha_t + 1 / \eta_t}$ , we get the update rule
+
+$$
+W _ {t} \leftarrow c \operatorname {S o f t m a x} \left(\left(1 - \lambda_ {t}\right) \log \left(W _ {t - 1}\right) - \eta_ {t} ^ {\prime} \nabla \ell \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right)\right) \tag {21}
+$$
+
+where $\lambda_t \in (0,1)$ and $\eta' \in \mathbb{R}^+$ are the parameters that can be learned during training. The Softmax operator ensures that the output lies in the set $\mathcal{W}$ .
+
+Notice that while all above calculations are done for a matrix $W$ , similar update rule holds for other forms of parameters such as when $W$ is a neural network (or when the parameter $W$ is normalized per slice).
+
+Variant 2: Elastic Net Regularization: Hard and Soft Forgetting. Elastic net is a powerful and popular tool in regression analysis to balance the feature selection capabilities of LASSO (Tibshirani 1996) and bias reduction properties of Ridge regression (Hilt et al. 1977; Hoerl et al. 1970). It has been widely used in different applications due to its ability to handle high-dimensional data and mitigate the effects of multicollinearity. Given this success, a natural question is what happens if we use this regularization scheme in our context.
+
+Let us start based on (Learning-Retaining Viewpoint) to design our memorization scheme. In (Learning-Retaining Viewpoint), we discussed that the loss function $\widetilde{\ell_t} (W;\mathbf{k}_t,\mathbf{v}_t)$ is an approximation of the original function $\ell (\cdot)$ , measuring our goodness-of-fit. Regularizing this loss with elastic net regularizer, we obtain the approximation
+
+$$
+\widetilde {\ell} _ {t} (W; \mathbf {k} _ {t}, \mathbf {v} _ {t}) = \langle W - W _ {t - 1}, \nabla \ell (W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}) \rangle .
+$$
+
+with a global retention of $\mathrm{G}_t(W) = \frac{1}{2\beta} \| W\| _2^2 +\frac{1}{\alpha}\| W\| _1$ . To fully specify the update rule of (Learning-Retaining Viewpoint), we also need to specify the premetric functions $\mathrm{D}_t(\cdot ,\cdot)$ . For the sake of keeping the update rule simple (and parallelizable), we can choose
+
+$$
+\mathrm {D} _ {t} (W, W _ {t - 1}) = \frac {1}{2} \| W - W _ {t - 1} \| _ {2} ^ {2}.
+$$
+
+These choices of the attentional bias and retention gate leads to the following update rule:
+
+$$
+W _ {t} = \mathcal {S} _ {Y} \left(\lambda W _ {t - 1} - \zeta \nabla \ell \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right)\right), \tag {22}
+$$
+
+where $\gamma = \frac{\eta\beta}{\alpha(\eta + \beta)}$ , $\lambda = \frac{\beta}{\beta + \eta}$ , $\zeta = \eta\lambda$ , and $S_{\gamma}$ is the soft thresholding operator, applied element-wise. For each element, this operator is defined as
+
+$$
+\mathcal {S} _ {\gamma} (z) = \operatorname {s i g n} (z) \max \left\{0, | z | - \gamma \right\}.
+$$
+
+In other words, for large values of $z$ , $S_{\gamma}(z)$ makes $z$ closer to zero by $\gamma$ amount. If it is already in the $\gamma$ -vicinity of zero, then it makes it zero (hard forget).
+
+Equation (22) can be viewed as a combination of soft forgetting (obtained by multiplying $W$ by $\lambda \in (0,1)$ , and a hard forgetting (if it is smaller than $\gamma$ ). The hyperparameters $\gamma, \lambda,$ and $\zeta$ can be learned. Notice that since the shrinkage operator is not differentiable, we can approximate it with its smooth approximation. For example, we can use $S_{\gamma}(z) \approx \frac{|z|*\arctan(z / \gamma)}{\pi / 2}$ .
+
+Variant 3: Elastic Net Regularization: Forgetting via Soft-thresholding. The elastic net regularizer can also be used in the (FTRL Viewpoint). In particular, in (FTRL Viewpoint), we can set
+
+$$
+\frac {1}{\eta_ {t}} R _ {t} (W) = \frac {1}{\eta} \| W \| ^ {2} + \frac {1}{\alpha} \| W \| _ {1}
+$$
+
+and use $\widehat{\ell}(W; x_i) = \langle W - W_{i-1}, \nabla \ell(W_{i-1}; x_i) \rangle$ . Assuming initialization at $W_0 = 0$ , these choices of attentional bias and retention gate leads to the update rules:
+
+$$
+A _ {t} = A _ {t - 1} - \eta \nabla \ell \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right)
+$$
+
+$$
+W _ {t} = \mathcal {S} _ {\eta / \alpha} (A _ {t}) \tag {23}
+$$
+
+Here $S_{\eta /\alpha}(\cdot)$ is the soft-thresholding operator with parameter $\eta /\alpha$ , which can be smoothly as explained in Variant 1.1.
+
+Variant 4: General $L_{q}$ Memory Stability. Existing work is based on the retention gate choices $\mathrm{D}_t(W, W_{t-1}) = \|W - W_{t-1}\|_F^2$ or $R(W) = \|W\|_2^2$ . However, one can choose other choices of retention gate. For example, in (FTRL Viewpoint), we can choose $L_{q}$ norm as the regularizer $R(W)$ . More specifically, for $1 < q \leq 2$ , we can set
+
+$$
+\frac {1}{\eta_ {t}} R (W) = \frac {1}{2 \eta (q - 1)} \| W \| _ {q} ^ {2}.
+$$
+
+Using this retention gate and choosing $\widehat{\ell_i} (W;\mathbf{k}_t,\mathbf{v}_t) = \langle W - W_{i - 1},\nabla \ell (W_{i - 1};\mathbf{k}_t,\mathbf{v}_t)\rangle$ in (FTRL Viewpoint), leads to the update rule $W_{t} = -\eta \frac{A_{t}}{\|A_{t}\|_{p}^{p - 2}}$ , where $p = \frac{q}{q - 1}$ and $A_{t} = \sum_{i = 1}^{t}\nabla \ell (W_{i - 1};\mathbf{k}_{t},\mathbf{v}_{t})$ ; see Shalev-Shwartz et al. 2012, Section 2.6. Here, $\odot$ denotes the Hadamard (element-wise) product and $|\cdot |$ is the element-wise absolute value operator. Assuming $W_0 = 0$ , this update rule can be recursively written as:
+
+$$
+A _ {t} = A _ {t - 1} - \eta \nabla \ell \left(W _ {i - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right), \quad \text {a n d} \quad W _ {t} = \frac {A _ {t}}{\| A _ {t} \| _ {p} ^ {p - 2}}.
+$$
+
+Variant 5: Bregman Divergence as Retention Gate.. Another natural choice is to use Bregman divergence as retention gate, leading to a mirror descent-type algorithms. In particular, given a smooth strictly convex function $f(\cdot): \mathbb{R} \mapsto \mathbb{R}$ , we can define the function $F(W) = \sum_{jl} f(W_{jl})$ . Based on this choice of function $F$ , we define the Bregman divergence
+
+$$
+D _ {t} (W, W ^ {\prime}) = F (W) - F \left(W ^ {\prime}\right) - \langle W ^ {\prime}, W - W ^ {\prime} \rangle
+$$
+
+as our parametric function. Utilizing this retention gate and choosing $\widetilde{\ell}_t(W;\mathbf{k}_t,\mathbf{v}_t) = \langle W - W_{t - 1},\nabla \ell (W_{t - 1};\mathbf{k}_t,\mathbf{v}_t)\rangle$ in (Learning-Retaining Viewpoint), we obtain the update rule
+
+$$
+W _ {t} = g \left(- \eta \nabla \ell \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right) + F ^ {\prime} \left(W _ {t - 1}\right)\right).
+$$
+
+Here, $F'$ is the mapping obtained by applying $f'(\cdot)$ (the derivative of $f$ ) element-wise to all entries of its input matrix argument. The function $g$ is the inverse of the mapping $F'(\cdot)$ , i.e., $g(F'(W)) = W$ .
+
+If we choose $f(\tau) = \frac{\tau^2}{2}$ , then $F'(W)$ becomes the identity mapping and so is $g$ . Therefore, the above update becomes simple gradient descent with no nonlinearity involved in the update rule. However, other choices of $f(\cdot)$ introduces additional nonlinearity in $g(\cdot)$ , which can enhance the expressivity of our memory. For example, we can choose the function $f(\cdot)$ so that its derivative becomes the inverse sigmoid function, i.e., $f'(\tau) = \ln \left( \frac{\tau}{1 - \tau} \right)$ with $f': (0,1) \mapsto \mathbb{R}$ . Since $f'(\cdot)$ is strictly increasing, then the function $f(\cdot)$ (and hence $F(\cdot)$ ) is strictly convex. Therefore, the Bregman divergence is well defined. Moreover, the inverse of the function $f'(\cdot)$ becomes the sigmoid function, i.e., $g(\tau) = \sigma(\tau) = \frac{\exp(\tau)}{1 + \exp(\tau)}$ with $g: \mathbb{R} \mapsto (0,1)$ . Then, the update of the memory becomes
+
+$$
+W _ {t} = \sigma \left(\ln \left(\frac {W _ {t}}{1 - W _ {t}}\right) - \eta \nabla \ell (W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t})\right),
+$$
+
+where $\sigma$ is the sigmoid function operated element-wise on the entries of $W$ , and the division operator $\frac{W_t}{1 - W_t}$ is also performed element-wise. This update rule guarantees that the elements of $W_t$ remain within the interval $(0, 1)$ .
+
+# 5.3 MIRAs's Variants: MONETA, YAAD, and MEMORA
+
+In the previous section we discussed different potential choices for attentional bias and retention gate to show the generality and the potential of MIRAs. In this section, building upon our framework, we present three novel sequence models, each of which designed based on a different motivation, and discuss how they can leverage fast parallel training.
+
+MOnETA. Given $p,q\in \mathbb{R}^{\geq 1}$ , we design $(p,q)$ -MONETA as the variant of MIRAs as follows: (1) For the choice of memory architecture, we use an MLP with 2 layers with expansion factor of 4 and GELU activation function (Hendrycks et al. 2016). We also use residual connections and layer norm, resulting in $\mathcal{M}(x) = x + \mathsf{LN}(W_1\sigma (W_2x))$ . (2) We choose $\ell_p$ -attentional bias (introduced in Equation 11) for MONETA. (3) For the choice of retention gate, we use the hybrid of $\ell_q$ retention gate $\frac{1}{2(q - 1)}\| W\| _q^2$ (see Section 5.2 for details) and the standard $\ell_2$ regularization $\frac{1}{\beta}\| W\| _2^2$ . (4) Finally, we use gradient descent as the memory learning algorithm. The above choices, result in the following recurrent formula for the memory module:
+
+$$
+A _ {t} = \alpha_ {t} A _ {t - 1} - \eta_ {t} \nabla \ell_ {p} \left(W _ {i - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right), \quad \text {a n d} \quad W _ {t} = \frac {A _ {t}}{\| A _ {t} \| _ {q} ^ {q - 2}}. \tag {24}
+$$
+
+Notably the gradient can be calculated using:
+
+$$
+\nabla \ell \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right) = p \eta_ {t} \left(\operatorname {S i g n} \left(W \mathbf {k} _ {t} - \mathbf {v} _ {t}\right) \odot | W \mathbf {k} _ {t} - \mathbf {v} _ {t} | ^ {p - 1}\right) \mathbf {k} _ {t} ^ {\top}. \tag {25}
+$$
+
+We use $(p,q) = (3,4)$ .
+
+YAAD. Building upon our discussion on the importance of robust memory that protects itself from extreme events (tokens), we design YAAD based on Huber objective. That is, in MirAS, for the choice of memory structure, we follow MONETA and use an MLP with the same architecture as above; for the choice of attentional bias, we use Huber loss (defined in Equation 16); for the choice retention gate, for the sake of simplicity, we use a combination of local and global retention as $\mathrm{Ret}_t(W,W_{t - 1}) = \frac{1}{2\theta_t}\| W - W_{t - 1}\| _F^2 +\frac{1}{\beta_t}\| W\| _2^2$ , which is equivalent to the "forget gate" mechanism introduced by Behrouz et al. (2024c); and finally, we simply use gradient descent as the memory learning algorithm. Given the above choices, we can write the resulted memory learning process as follows:
+
+$$
+W _ {t} = \alpha_ {t} W _ {t - 1} - \left\{ \begin{array}{l l} \eta_ {t} \nabla \ell_ {2} \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right) & \text {i f} \| \mathcal {M} (\mathbf {k} _ {t}) - \mathbf {v} _ {t} \| \leq \delta_ {t}, \\ \eta_ {t} \delta_ {t} \nabla \ell_ {1} \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right) & \text {O t h e r w i s e .} \end{array} \right. \tag {26}
+$$
+
+Note that for improving the expressive power, in all architectures, we decouple the learning rate $\eta$ and the retention gate rate $\alpha$ , resulting in an independent parameter $\beta_{t} \in [0,1]^{d}$ .
+
+
+Figure 2: Visualization of the MirAs's variant architecture, their hybrid counterpart with SWA, and block design of MirAs layer.
+
+
+
+
+
+MEMORA. Finally, in MEMORA, we use the idea of elastic net regularization (i.e., hard and soft retention). To this end, in Miras: (1) For the choice of memory architecture, similar to above variants, we use an MLP (the same architecture as the previous variants). (2) For the choice of attentional bias, we use simple $\ell_2$ regression loss. (3) For the choice of retention gate we use KL divergence as in Equation 21. (4) Finally, we optimize the memory using gradient descent, resulting in the following update rule:
+
+$$
+W _ {t} = \operatorname {S o f t m a x} \left(\alpha_ {t} \log \left(W _ {t - 1}\right) - \eta_ {t} \nabla \ell_ {2} \left(W _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right)\right) \tag {27}
+$$
+
+# 5.4 Architecture Backbone and Fast Training
+
+Architectural Backbone. For the architectural backbone, we fully follow recent studies (Behrouz et al. 2024c; Yang et al. 2024a): We replace attention modules with our variants of MIRAs in Llama's macro architecture with MLPs with SwiGLU(. ) activation, rotary positional encodings (RoPE) (Su et al. 2024), and RMSNorm (Zhang et al. 2019). For MIRAs layer block, we follow the recent modern linear recurrent models (Behrouz et al. 2024c; Yang et al. 2024a), and incorporate a 1D depthwise-separable convolution layer (with kernel size of 4) after each of the query, key, and value projections. For the sake of training stability, we also use $\ell_2$ normalization to $\mathbf{q}$ and $\mathbf{k}$ . The output of MIRAs layer block is normalized and gated with a linear layer (Mehta et al. 2023).
+
+Channel-wise Parameters. For learnable parameters of $\eta_t, \delta_t$ and the retention gate of $\alpha_t$ we use channel-wise parametrization, i.e., $\eta_t, \delta_t, \alpha_t \in \mathbb{R}^d$ . While gaining more expressive power, this parametrization results in significant parameter increase. To mitigate this issue, following Peng et al. (2025b), we use low-rank projections to project the input into $\mathbb{R}^k$ and then to $\mathbb{R}^d$ , where $k$ is a hyperparameter (usually 32 or 64). The backbone architecture is illustrated in Figure 2.
+
+Hybrid Models. We also evaluate the hybrid version of Miras's variants. For hybrid models, we follow the Samba (Ren et al. 2024) architecture, in which we sequentially combine our Miras layer with Sliding Window Attention (SWA). The illustration of hybrid model Figure 2.
+
+Parallelizable Training. While the design of Miras's variant are theoretically well-motivated, their recurrence is non-linear, potentially making their straightforward training slow for large scales. In this section, we build upon the work of Behrouz et al. (2024c) and Sun et al. (2024) to make the training parallelizable. The main idea is to divide the sequence into
+
+chunks with size $b$ (usually is 16 or 64) and calculate the gradient for all tokens in the current chunk with respect to the last state of the memory in the previous chunk. That is, we use $\nabla \ell(\mathcal{M}_{t'}; \mathbf{k}_t, \mathbf{v}_t)$ instead of $\nabla \ell(\mathcal{M}_{t-1}; \mathbf{k}_t, \mathbf{v}_t)$ , where $t'$ is the last state in the previous chunk.
+
+Given the above trick, we can calculate all gradients at once and make the recurrence inside each chunk linear. However, to fully take advantage of accelerators, we need to reformulate the process as matrix multiplication. For MONETA, for the sake of clarity, assume $q = 2$ . We follow the same algorithm as Behrouz et al. (2024c) and expand the recurrence as follows:
+
+$$
+\begin{array}{l} \mathcal {M} _ {t} = \alpha_ {t} \mathcal {M} _ {t - 1} - \eta_ {t} \nabla \ell (\mathcal {M} _ {t - 1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}) \\ = \beta_ {t} \mathcal {M} _ {0} - \sum_ {i = 1} ^ {t} \eta_ {i} \frac {\beta_ {t}}{\beta_ {i}} \nabla \ell \left(\mathcal {M} _ {t ^ {\prime}}; \mathbf {k} _ {i}, \mathbf {v} _ {i}\right), \tag {28} \\ \end{array}
+$$
+
+where $t' = t - \mathrm{mod}(t, b)$ , and $\beta_{i} = \prod_{j=1}^{i} \alpha_{j}$ . For the sake of clarity, we focus on the first chunk, i.e., $t = b$ and so $t' = 0$ , and explain the process for the case that $\mathcal{M}_t = W_t$ is linear. The process for 2-layer MLPs and other chunks is similar. Using $\ell_p$ loss function, we have:
+
+$$
+\begin{array}{l} \nabla \ell \left(W _ {0}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right) = p \left(\operatorname {S i g n} \left(W _ {0} \mathbf {k} _ {t} - \mathbf {v} _ {t}\right) \odot \left| W _ {0} \mathbf {k} _ {t} - \mathbf {v} _ {t} \right| ^ {p - 1}\right) \mathbf {k} _ {t} ^ {\top} \\ \Rightarrow \sum_ {i = 1} ^ {b} \eta_ {i} \frac {\beta_ {b}}{\beta_ {i}} \nabla \ell \left(W _ {0};; \mathbf {k} _ {i}, \mathbf {v} _ {i}\right) = p \mathbf {E} _ {b} \odot \mathbf {B} _ {b} \odot \operatorname {S i g n} \left(W \mathbf {k} _ {t} - \mathbf {v} _ {t}\right) \odot \left(\left| W _ {0} \mathbf {K} - \mathbf {V} \right| ^ {p - 1}\right) \mathbf {K} ^ {\top}, \tag {29} \\ \end{array}
+$$
+
+where $\mathbf{E}_b = \left[\eta_1\quad \eta_2\quad \dots \quad \eta_b\right]$ and $\mathbf{B}_b$ is defined analogously on $\frac{\beta_b}{\beta_i}\mathrm{s}$ . For the sake of stability in training, we use $\operatorname{Sign}(x)\approx \tanh (\alpha x)$ and $|x| = \sqrt{x^2 + \epsilon}$ , where $\epsilon >0$ is a small number (i.e., $\epsilon = 1e - 6$ ). As discussed in Equation 24, the case that $q\neq 2$ appears as a normalization term on the memory. Similar to Titans (Behrouz et al. 2024c) and TTT (Sun et al. 2024), we do not apply this non-linearity inside each chunk and instead use it at the end of each chunk.
+
+For YAAD, the process is very similar to the above. We calculate the gradient of both $\ell_1$ and $\ell_2$ loss and use a masking based on $\| \mathcal{M}(\mathbf{k}_t) - \mathbf{v}_t\| \leq \delta_t$ .
+
+For MEMORA, the update rule has two non-linear parts, i.e., softmax and log, making the model hardly parallelizable. To this end, as discussed above, we use its linear version inside each chunk and its non-linear version across chunks. However, using both log and softmax at the end of each chunk removes the effect of log. To this end, we consider a lag tokens after each chunk (i.e., tokens with index $i = kb + 1$ , where $b$ is the chunk size and $k \in \mathbb{Z}^+$ ). That is, let $\mathcal{M}_0$ be the last state of the memory in previous chunk, we have:
+
+$$
+\mathcal {M} _ {1} = \operatorname {S o f t m a x} \left(\alpha_ {1} \log \left(\mathcal {M} _ {0}\right) - \eta_ {1} \nabla \ell_ {2} \left(\mathcal {M} _ {0}; \mathbf {k} _ {1}, \mathbf {v} _ {1}\right)\right), \tag {30}
+$$
+
+and then we use $\mathcal{M}_1$ for the next chunk. Again, for the sake of clarity, assume that memory is linear, i.e., $\mathcal{M}_1 = W_1$ :
+
+$$
+\begin{array}{l} \nabla \ell \left(W _ {1}; \mathbf {k} _ {t}, \mathbf {v} _ {t}\right) = \left(W _ {1} \mathbf {k} _ {t} - \mathbf {v} _ {t}\right) \mathbf {k} _ {t} ^ {\top} (31) \\ \Rightarrow \sum_ {i = 1} ^ {b} \eta_ {i} \frac {\beta_ {b}}{\beta_ {i}} \nabla \ell \left(W _ {1};; \mathbf {k} _ {i}, \mathbf {v} _ {i}\right) = \mathbf {E} _ {b} \odot \mathbf {B} _ {b} \odot \left(W _ {1} \mathbf {K} - \mathbf {V}\right) \mathbf {K} ^ {\top}, (32) \\ \end{array}
+$$
+
+where matrices are defined the same as for Equation 29.
+
+# 6 Experiments
+
+In our experimental evaluations, we aim to answer three main questions: (1) Does different attentional biases results in different architectures in practice? (2) How does different types of retention gates (i.e., retention gate) affect the performance of the model in long context? (3) How do MEMORA, MONETA, and YAAD perform in downstream tasks compare to baselines?
+
+Setup. We train our models with training context window of size 4096 using either FineWeb-Edu dataset (Penedo et al. 2024) (for LM and common-sense reasoning tasks) or C4 dataset (Raffel et al. 2020) (for scaling patterns). We use model
+
+
+Figure 3: Scaling patterns when increasing (Left) model size, (Middle) sequence length (model size = 340M) (3) (Right) sequence length (model size = 760M) on C4 dataset.
+
+
+
+
+
+sizes of 120M, 340M, 760M, and 1.3B parameters. We train small models (120M and 340M) on 15B tokens sampled from the dataset, the medium size model (760M) on 30B tokens, and the large model on 100B tokens. Baseline results are reported by Behrouz et al. (2024c).
+
+# 6.1 Language Modeling and Common-sense Reasoning
+
+We follow recent studies (Behrouz et al. 2024c; Yang et al. 2024a,c) and first focus on the perplexity in language modeling and also commonsense reasoning tasks. The results for MEMORA, YAAD, MONETA and also baselines with size of 340M, 760, and 1.3B are reported in Table 2. All of our variants outperforms all the baselines including Transformer++, modern linear recurrent models and hybrid methods. The superior performance compared to hybrid models is particularly important as all of our variants are pure recurrent (attention-free). Among the three variants of MirAS, while MONETA achieves slightly weaker performance than MEMORA, and YAAD, the other two variants are close and depending on the task and model size, the best model can vary.
+
+# 6.2 Scaling Pattern
+
+To evaluate the scaling pattern of models and for comparing them with baseline, in this section, we plot their performance with varying the model size and the context window.
+
+Context Length. We first vary the training context length from 2K to 32K for two versions of our model with size 340M and 760M. The results are reported in Figure 3 (Middle and Right). All three variants of Miras scales better than state-of-the-art baselines when increasing the context length. We attribute this superior performance to: (1) expressive memory architecture. Contrary to baselines like Mamba2 and GSA that uses vector- and matrix-valued memory, our variants are using 2-layer MLPs with more expressive power to learn from longer sequences. (2) The choice of retention gate and attentional bias: All of our three variants go beyond the standard attentional biases and retention gates. These choices can help the memory to better manage its fixed-size capacity.
+
+Model Size. We also report the #FLOPs vs. perplexity of our models and baselines in Figure 3 (Left). All three variants outperforms all baselines given almost the same budget of FLOPs. These results, once again support the importance of powerful memory design.
+
+# 6.3 Needle In Haystack
+
+To evaluate the effective context window of our models and baselines, we use needle-in-haystack task. In this task, we evaluate the model on retrieving a piece of information (i.e., the "needle") from long distractor texts (i.e., the "haystack"). We focus on the Single NIAH (S-NIAH) task from RULER benchmark (Hsieh et al. 2024) and evaluate our models and baselines on sequences with length 1K, 2K, 4K, and 8K. The results are reported in Table 3. All our variants outperforms all the baselines with a considerable margin. Interestingly, MONETA shows better performance than others when the data is synthetic noise (S-NIAH-PK). This observation validates the effectiveness of $p$ -norm objective and retention gates as they are more robust to noise.
+
+Table 2: Performance of MIRAS's variants and baselines on language modeling and common-sense reasoning tasks. Hybrid models are marked with *. The best results of simple and hybrid models are highlighted. In largest scale, we compare our simple models with even hybrid models and highlight the best results.
+
+| Model | Memory Architecture | Attentional Bias | Retention Gate† | Memory Algorithm | Memory Write Operation |
| Shallow Memory |
| RetNet (2023) | Vector | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| Transformer (2017) | Matrix | L2 | - | Nonparametric | Mt=Mt-1∪{kt, vt} |
| LA (2021) | Matrix | Dot-Product | - | GD | Mt=Mt-1+vtktT |
| DFW | Matrix | Dot-Product | L2 | GD | Mt=(βtαT) ⊙ Mt-1+vtktT |
| Lightening Attention (2025) | Matrix | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| GLA (2024) | Matrix | Dot-Product | L2 | GD | Mt=Diag(αt)Mt-1+vtktT |
| Mamba (2024) | Matrix | Dot-Product | L2 | GD | Mt=αMt-1+vtktT |
| HGRN2 (2024) | Matrix | L1 | L2 | GD | Mt=Diag(αt)Mt-1+vt(1-αt)T |
| DeltaNet (2017) | Matrix | L2 | - | GD | Mt=(I-βtktkT)Mt-1+βtvtktT |
| Longhorn (2024) | Matrix | L2 | - | Implicit GD | Mt=(I-βtktkT)Mt-1+(βt1+ktkβt)xtkT |
| TTT-Linear (2024) | Matrix | L2 | - | GD | Mt=Mt-1-η∇L(Mt-1, xt) |
| Gated DeltaNet (2024) | Matrix | L2 | L2 | GD | Mt=(αt(I-βtktkT))Mt-1+βtvtktT |
| RWKV-7 (2025) | Matrix | L2 | L2 | GD | Mt=diag(αt)(I-βtktkT)Mt-1+βtvtktT |
| DeltaProduct (2025) | Matrix | L2 | L2 | MGD* | Mt=(αtΠi=1n(I-βt,ikt,i)T)Mt-1+Σj=1nΠi=j(I-βt,ivtj,kj,i) |
| Deep Memory |
| TTT-MLP (2024) | 2-layer MLP | L2 | - | GD | Mt=Mt-1-η∇L(Mt-1;kt, vt) |
| Titans-LMM (2024) | k-layer MLP | L2 | L2 | GD + Momentum | Mt=αMt-1-St, where St=ηSt-1-θt∇L(Mt-1;kt, vt) |
| MONETA (ours) | 2-layer MLP | Lp | Lq | GD | At=AtA1-ηt∇lp(Wt-1;kt, vt), Wt=At/||At||q-2 |
| YAAD (ours) | 2-layer MLP | Huber | L2 | GD | Wt=atWt-1-(ηt∇ε2(Wt-1;kt, vt) if ||M(kt)-vt|≤δt, ηtδt∇ε1(Wt-1;kt, vt) Otherwise. |
| MEMORA (ours) | 2-layer MLP | L2 | KL | GD | Wt=Softmax(αt log(Wt-1)-ηt∇ε2(Wt-1;kt, vt)) |
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+ "content": "For the sake of clarity, we use L2 for all modified L2-like regularizations. However, in fact, only Titans and RWKV-7 are using L2 retention gate (see Section 4)"
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+ "content": "Deep Memory Module: Titans and Test Time Training. To overcome the limited memory and to extend the effective context length of deep sequence models, more recent studies focus on a new generation of architectures with deep memory module (Behrouz et al. 2024c; Sun et al. 2024). These architectures are built on the meta-learning perspective, where the memory is an MLP architecture that is updated using gradient descent (with momentum) (Behrouz et al. 2024c; Sun et al. 2024). Sun et al. (2024) further provide a unifying perspective that how linear and softmax attention are respectively parametric and non-parametric solutions of (kernel) regression loss but consider other modern linear RNNs outside of this class of models. Recently, in a concurrent work to ours, Wang et al. (2025) show that with additional simplification of modern RNNs (e.g., RetNet (Sun et al. 2023), Mamba (Dao et al. 2024)) they approximately place in the same class of models that internally optimize regression loss. It, however, still remains unanswered that \"What is the underlying design framework behind these sequence models that can accurately unify existing architectures?\" Moreover, the role of forget gates and its alternative choices in modern sequence models is surprisingly less explored."
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+ "content": "W _ {t} = \\arg \\min _ {W \\in \\mathcal {W}} \\underbrace {\\sum_ {i = 1} ^ {t} \\widehat {\\ell_ {i}} (W ; \\mathbf {k} _ {i} , \\mathbf {v} _ {i})} _ {\\text {A t t e n t i o n a l B i a s}} + \\underbrace {\\frac {1}{\\eta_ {t}} \\mathcal {R} _ {t} (W)} _ {\\text {M e m o r y S t a b i l i t y}}. \\tag {FTRLViewpoint}",
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+ "content": "W _ {t} = \\arg \\min _ {W \\in \\mathcal {W}} \\underbrace {\\widetilde {\\ell_ {t}} (W ; \\mathbf {k} _ {t} , \\mathbf {v} _ {t})} _ {\\text {A t t e n t i o n a l B i a s}} + \\underbrace {\\operatorname {R e t} _ {t} (W , W _ {t - 1})} _ {\\text {R e t e n t i o n}}, \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\text {(L e a r n i n g - R e t a i n i n g V i e w p o i n t)}",
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+ "content": ", aims at retaining previously learned knowledge. The coefficient "
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+ "content": " can be viewed as a meta in-context learning rate, where larger values of "
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+ "type": "text",
+ "content": " leads to learning more from new concepts, while allowing higher forgetting of previously learned concepts. The second term is a global retention that controls the change of the memory with respect to its size. The special instances of the above viewpoint (e.g., without global retention, with implicit closed-form solution, and/or with limited memory structure) have been the motivation behind some of the recent studies such as Liu et al. (2024a)."
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+ "content": "The (FTRL Viewpoint) and (Learning-Retaining Viewpoint) are connected through the lens of online optimization. For example, as discussed above, by choosing linear approximation of the loss and quadratic regularization/retention, they can both cover online gradient descent update in (5) as a special case. One straightforward way to make the connection explicit is by defining the premetric "
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+ "type": "text",
+ "content": " based on the previous loss functions and the regularization, as described in Proposition 3.2 below:"
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+ "content": "\\mathcal{W} = \\mathbb{R}^d"
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+ "content": " in (Learning-Retaining Viewpoint). Then, the update rule in (Learning-Retaining Viewpoint) is equivalent to the update rule in (FTRL Viewpoint)."
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+ "content": " in (Learning-Retaining Viewpoint)). In parametric setups, this choice balances learning with retention of past state. An effective retention gate is key to the good performance in long context tasks."
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+ "content": "\\mathrm{Ret}_t(\\mathcal{M}, \\mathcal{M}_{t-1}) = \\| \\mathcal{M}_t - \\alpha \\mathcal{M}_{t-1} \\|_F^2"
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+ "content": ". Using Equation Learning-Retaining Viewpoint and gradient descent as the optimizer (i.e., memory learning algorithm), the memory update rule is:"
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+ "content": "\\mathcal {M} _ {t} = \\alpha \\mathcal {M} _ {t - 1} + \\mathbf {v} _ {t} \\mathbf {k} _ {t} ^ {\\top}. \\tag {8}",
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+ "content": "RNNs with Delta Rule. To improve the memory management and to enhance the memory capacity of the above group, several studies suggest using delta rule (Neil et al. 2017; Schlag et al. 2021) as the learning algorithm in recurrent neural networks (e.g., DeltaNet (Schlag et al. 2021), Longhorn (Liu et al. 2024a), and RWKV7 (Peng et al. 2025b)). In this part, we recall that where "
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+ "content": "\\mathrm{Ret}_t(\\mathcal{M}, \\mathcal{M}_{t-1}) = \\| \\mathcal{M}_t - \\alpha \\mathcal{M}_{t-1} \\|_F^2"
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+ "content": "\\mathcal {M} _ {t} = \\alpha \\left(\\mathbf {I} - \\eta_ {t} \\mathbf {k} _ {t} \\mathbf {k} _ {t} ^ {\\top}\\right) \\mathcal {M} _ {t - 1} + \\mathbf {v} _ {t} \\mathbf {k} _ {t} ^ {\\top}. \\tag {9}",
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+ "type": "text",
+ "content": ", memory update is equivalent to DeltaNet (Schlag et al. 2021); and (2) "
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+ "content": " are data-dependent learnable parameters, resulting sequence model is either Gated DeltaNet (Yang et al. 2024a) ("
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+ "type": "text",
+ "content": "), or RWKV7 (Peng et al. 2025b) ("
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+ "type": "text",
+ "content": "). Therefore, RNNs with delta rule are special instances of MIRAS."
+ }
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+ }
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+ },
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+ "content": "Beyond Delta Rule. As discussed earlier, while delta rule with its value replacement strategy is more powerful than Hebbian-like learning rules, it suffers from theoretical limitations (Irie et al. 2023) and achieves moderate performance in practice (Yang et al. 2024c). Therefore, several studies have focused on update rules beyond delta rule. Recently, Titans (Behrouz et al. 2024c) suggests using non-linear MSE objective of "
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+ "type": "inline_equation",
+ "content": "\\| \\mathcal{M}_t(\\mathbf{k}_t) - \\mathbf{v}_t\\| _2^2"
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+ "content": "\\mathrm{D}_t = \\| W_t - W_{t - 1}\\| _F^2"
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+ "type": "text",
+ "content": ". Therefore, Titans-LMM is a special instance of MIRAs, where we use the abovementioned attentional bias and retention regularizations, and gradient descent with momentum as the optimizer."
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+ "type": "inline_equation",
+ "content": "\\sum_{i=1}^{t} \\|\\mathcal{M}_{t}(\\mathbf{k}_{i}) - \\mathbf{v}_{i}\\|_{2}^{2}"
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+ "type": "text",
+ "content": " as the retention regularization. Since these models use Newton's method to optimize such an objective, they provide a more expressive update rule than delta rule. We further discuss a set of new learning algorithms beyond delta rule in Section 5."
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+ "type": "text",
+ "content": ") with Nadaraya-Watson estimator. Therefore, softmax attention is an instance of MIRAS, when we find the non-parametric solution to the MSE loss with Nadaraya-Watson estimator, without retention."
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+ "content": "5 Beyond Existing Attentional Biases and Retention Gates"
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+ "content": "L(\\mathcal{M}(\\mathbf{k}_t),\\mathbf{v}_t) = c_t\\| \\mathcal{M}(\\mathbf{k}_t) - \\mathbf{v}_t\\|^2"
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+ "content": " is considered as a quadratic/linear function. In addition, almost all prior work considers "
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+ "content": " to be the entire "
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+ "type": "text",
+ "content": " space. However, in general there could be various choices for all the three aforementioned design choices. To illustrate the potential and flexibility of our designed framework, here, we review some of the potential design choices for attentional bias and retention gate in MirAS. For the sake of clarity, we discuss all these attentional bias and memory retention gates based on using gradient descent as the optimizer, and so based on the provided two view points. However, these attentional bias objectives and retention regularizers can be directly used in Equation 4 and optimized by using any other optimization algorithms, resulting in different update rules."
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+ "content": "-Attentional Bias. As discussed in the main body, attentional bias defines the \"similarity metric\" and measures how well memory can recall the value, given its corresponding key. Although "
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+ "type": "text",
+ "content": "-attentional bias as:"
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+ "content": "\\mathcal {L} \\left(\\mathcal {M} \\left(W, \\mathbf {k} _ {t}\\right); \\mathbf {v} _ {t}\\right) = \\| \\mathcal {M} \\left(\\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {p} ^ {p}, \\tag {10}",
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+ "content": "The retention gate (forget gate) in Titans is different from Mamba2 and Gated DeltaNet that we discussed above. The main difference comes from the case of full memory erase. While Mamba2 gating removes the entire memory and treats the next token as the first ever seen data, Titans use a \"cold start\" strategy and use the previous state of the memory to measure the surprise of the incoming token before fully erasing the memory."
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+ "content": " are non-differentiable and so might cause unstable training. To overcome this issue, we use "
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+ "type": "interline_equation",
+ "content": "W _ {t} = W _ {t - 1} - \\eta_ {t} \\left[ \\left(\\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {T}\\right) \\odot \\left(\\mathbf {I} \\left(\\left| W \\mathbf {k} _ {t} - \\mathbf {v} _ {t} \\right| \\leq \\delta_ {t}\\right) \\mathbf {1} ^ {\\top}\\right) + \\left(\\delta_ {t} \\operatorname {S i g n} \\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\mathbf {k} ^ {\\top}\\right) \\odot \\left(\\mathbf {I} \\left(\\left| W \\mathbf {k} _ {t} - \\mathbf {v} _ {t} \\right| > \\delta_ {t}\\right) \\mathbf {1} ^ {\\top}\\right) \\right] \\tag {14}",
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+ "content": "W _ {t} = W _ {t - 1} - \\eta_ {t} \\left\\{ \\begin{array}{l l} \\left(\\mathcal {M} \\left(W _ {t - 1}, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {T} & \\text {i f} \\| \\mathcal {M} \\left(W _ {t - 1}, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2} \\leq \\delta_ {t}, \\\\ \\delta_ {t} \\frac {\\left(\\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}\\right)}{\\| \\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2}} \\mathbf {k} _ {t} ^ {T} & \\text {O t h e r w i s e .} \\end{array} \\right. \\tag {15}",
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+ "content": "W _ {t} = W _ {t - 1} - \\left\\{ \\begin{array}{l l} \\eta_ {t} \\nabla \\ell_ {2} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) & \\text {i f} \\| \\mathcal {M} (\\mathbf {k} _ {t}) - \\mathbf {v} _ {t} \\| \\leq \\delta_ {t}, \\\\ \\eta_ {t} \\delta_ {t} \\nabla \\ell_ {1} \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) & \\text {O t h e r w i s e .} \\end{array} \\right. \\tag {16}",
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+ "content": "Variant 3: Memory Robust to Value Shifts. Following the robustness requirement discussed in the previous section, we aim to design a memory mechanism that exhibits resilience against small shifts in the value parameter. A natural approach in this context is to employ a robust optimization formulation. Specifically, we define the loss function as the worst-case "
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+ "content": "\\delta \\mathbf {v} _ {t} ^ {*} = \\Delta \\frac {- \\mathcal {M} (W , \\mathbf {k} _ {t}) + \\mathbf {v} _ {t}}{\\| \\mathcal {M} (W , \\mathbf {k} _ {t}) - \\mathbf {v} _ {t} \\| _ {2}}",
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+ "content": " thus controls the trade-off between fitting the nominal data and ensuring robustness against value perturbations."
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+ "type": "inline_equation",
+ "content": "\\widetilde{\\ell} (W;\\mathbf{k}_t,\\mathbf{v}_t) = \\langle W - W_{t - 1},\\nabla \\ell (W_{t - 1};\\mathbf{k}_t,\\mathbf{v}_t)\\rangle"
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+ "content": ". Furthermore, to simplify the memory operation, we assume a linear matrix memory model "
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+ "type": "inline_equation",
+ "content": "\\mathcal{M}(W,\\mathbf{k}_t) = W\\mathbf{k}_t"
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+ "type": "text",
+ "content": ". Under these assumptions, we can derive the memory update mechanism using gradient descent on the robust loss:"
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+ "type": "interline_equation",
+ "content": "W _ {t} = W _ {t - 1} - \\eta \\left(\\left(\\mathcal {M} \\left(W _ {t - 1}, \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}\\right) \\mathbf {k} _ {t} ^ {\\top} + \\Delta \\frac {\\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t}}{\\| \\mathcal {M} \\left(W _ {t - 1} , \\mathbf {k} _ {t}\\right) - \\mathbf {v} _ {t} \\| _ {2}} \\mathbf {k} _ {t} ^ {\\top}\\right)",
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+ "content": "In this update rule, the parameter "
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+ "type": "text",
+ "content": ", which governs the influence of the robustness term, can also be treated as a learnable parameter, allowing the model to adapt its robustness based on the observed data."
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+ "type": "text",
+ "content": "5.2 Alternative Retention Gates"
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+ "content": "Variant 1: Memorization Over A Scaled Probability Simplex Via "
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+ "type": "text",
+ "content": "-Divergence. A common technique in learning to prevent numerical instabilities and exploding values is to restrict the search space to a bounded domain. Following this principle, to avoid numerical instabilities, we can constrained the variable "
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+ "type": "text",
+ "content": " to lie within a (scaled) probability simplex. In other words, we can restrict the state to lie in the constraint set"
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+ "content": "\\mathcal {W} = \\{W \\mid \\| W \\| _ {1} = c \\text {a n d} W _ {j l} \\geq 0, \\forall j, l \\}.",
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+ "content": "In this set, each matrix "
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+ "content": "W"
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+ {
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+ "type": "text",
+ "content": " can be viewed as a measure. Thus, in (Learning-Retaining Viewpoint), we can utilize divergences over measures to define our premetric. For example, we can use "
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+ "content": "-divergence measure (Polyanskiy et al. 2025, Def 4.9), (Csiszar 1967) to define "
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+ "type": "text",
+ "content": ". More specifically, let "
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+ "content": "f(\\cdot)"
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+ "content": " be a smooth strictly convex function from "
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+ "type": "text",
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+ "content": ". Then, we can define the "
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+ "content": "-divergence between "
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+ "type": "text",
+ "content": " as"
+ }
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+ "type": "interline_equation",
+ "content": "\\mathrm {D} _ {t} (W, W ^ {\\prime}) = \\sum_ {j l} W _ {j l} ^ {\\prime} f \\left(\\frac {W _ {j l}}{W _ {j l} ^ {\\prime}}\\right).",
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+ "content": "-divergence is zero if and only if "
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+ "type": "inline_equation",
+ "content": "W = W'"
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+ "type": "text",
+ "content": "; see Polyanskiy et al. 2025, Theorem 2.3. Using the above premetric as the retention gate and setting "
+ },
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+ "type": "inline_equation",
+ "content": "\\widetilde{\\ell}(W; \\mathbf{k}_t, \\mathbf{v}_t) = \\langle W - W_{t-1}, \\nabla \\ell(W_{t-1}; \\mathbf{k}_t, \\mathbf{v}_t) \\rangle"
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+ "type": "text",
+ "content": " in (Learning-Retaining Viewpoint), we get the update rule"
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+ }
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+ "type": "interline_equation",
+ "content": "W _ {t} = W _ {t - 1} \\odot g \\left(- \\zeta_ {t} - \\eta_ {t} \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right). \\tag {18}",
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+ "content": " is the inverse of the mapping "
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+ "content": "g(f'(\\tau)) = \\tau"
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+ "type": "inline_equation",
+ "content": "\\forall \\tau"
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+ "type": "text",
+ "content": "; the operator "
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+ "type": "inline_equation",
+ "content": "\\odot"
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+ "type": "text",
+ "content": " denotes the Hadamard (elementwise) product, and "
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+ "type": "inline_equation",
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+ "type": "text",
+ "content": " should be chosen such that "
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+ "type": "inline_equation",
+ "content": "\\| W_t\\|_1 = c"
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+ "type": "text",
+ "content": ". Notice that since the function "
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+ "type": "text",
+ "content": " (which is strictly increasing) and define "
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+ "type": "inline_equation",
+ "content": "f(\\tau) = \\mathrm{const} + \\int_{\\tau' = 0}^{\\infty}g^{-1}(\\tau')d\\tau'"
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+ "type": "inline_equation",
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+ "type": "text",
+ "content": ". Then the update rule in (18) can be interpreted by the "
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+ "type": "inline_equation",
+ "content": "f"
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+ ],
+ "type": "text",
+ "content": "-divergence regularization, as explained above. Therefore, one can directly choose a continuous monotonically increasing function "
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+ "type": "inline_equation",
+ "content": "g(\\cdot)"
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+ "type": "text",
+ "content": " and use (18) for memory update."
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+ "type": "text",
+ "content": "Specializing to KL divergence. Let us further make the above update rule explicit by using special function "
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+ "type": "inline_equation",
+ "content": "f"
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+ "type": "text",
+ "content": ". If we choose "
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+ "type": "inline_equation",
+ "content": "f(\\tau) = \\tau \\ln(\\tau)"
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+ "type": "text",
+ "content": ", then the "
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+ "type": "text",
+ "content": "-divergence becomes the widely used KL divergence measure "
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+ "type": "inline_equation",
+ "content": "D_t(W, W_{t-1}) = \\sum_{jl} W_{jl} \\log \\left( \\frac{W_{jl}}{(W_t)_{jl}} \\right)"
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+ ],
+ "type": "text",
+ "content": ". In addition, we can also utilize the Shannon entropy as the global retention by regularizing deviations from uniform distribution, i.e., "
+ },
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+ "type": "inline_equation",
+ "content": "G_t(W) = \\sum_{jl} W_{jl} \\log (W_{jl})"
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+ ],
+ "type": "text",
+ "content": ". Combining these choices of the local and global retention gates, we obtain the overall retention gate"
+ }
+ ]
+ }
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+ "type": "interline_equation",
+ "content": "\\operatorname {R e t} _ {t} (W, W _ {t - 1}) = \\frac {1}{\\eta_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(\\frac {W _ {j l}}{\\left(W _ {t}\\right) _ {j l}}\\right) + \\frac {1}{\\alpha_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(W _ {j l}\\right)",
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+ }
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+ ],
+ "type": "text",
+ "content": "Choosing the attentional bias "
+ },
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+ ],
+ "type": "inline_equation",
+ "content": "\\widetilde{\\ell}(W; \\mathbf{k}_t, \\mathbf{v}_t) = \\langle W - W_{t-1}, \\nabla \\ell(W_{t-1}; \\mathbf{k}_t, \\mathbf{v}_t) \\rangle"
+ },
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+ "type": "text",
+ "content": " and the above retention gate will lead to the update rule"
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+ }
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+ },
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+ "type": "interline_equation",
+ "content": "W _ {t} = \\arg \\min _ {W} \\left\\langle W - W _ {t - 1}, \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) \\right\\rangle + \\frac {1}{\\eta_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(\\frac {W _ {j l}}{\\left(W _ {t}\\right) _ {j l}}\\right) + \\frac {1}{\\alpha_ {t}} \\sum_ {j l} W _ {j l} \\log \\left(W _ {j l}\\right) \\tag {19}",
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+ }
+ ],
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+ "type": "interline_equation",
+ "content": "\\text {s . t .} \\quad \\sum_ {j l} W _ {j l} = c, W _ {j l} \\geq 0, \\forall j l \\tag {20}",
+ "image_path": "eedbb2a68d612ba06fdfdba58b8ba19dcd6af7d8c89ad2b80e6e9ffecb747f21.jpg"
+ }
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+ "type": "text",
+ "content": "Attaching the Lagrange multiplier to the first constraint, the KKT conditions imply"
+ }
+ ]
+ }
+ ],
+ "index": 8
+ },
+ {
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+ "type": "interline_equation",
+ "content": "\\left(\\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right) _ {j l} + \\left(\\frac {1}{\\eta_ {t}} + \\frac {1}{\\alpha_ {t}}\\right) \\left(1 + \\log W _ {j l}\\right) - \\frac {1}{\\eta_ {t}} \\log \\left(\\left(W _ {t - 1}\\right) _ {j l}\\right) + \\mu_ {t} = 0, \\quad \\forall j, l",
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+ }
+ ]
+ }
+ ],
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+ "type": "text",
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+ "type": "text",
+ "content": "where "
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+ "type": "inline_equation",
+ "content": "\\mu_t"
+ },
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+ ],
+ "type": "text",
+ "content": " should be chosen such that "
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+ {
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+ "type": "inline_equation",
+ "content": "\\sum_{jl} W_{jl} = c"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ". Rearranging the terms and defining "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\lambda_t = \\frac{1 / \\alpha_t}{1 / \\alpha_t + 1 / \\eta_t}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\eta_t' = \\frac{1}{1 / \\alpha_t + 1 / \\eta_t}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", we get the update rule"
+ }
+ ]
+ }
+ ],
+ "index": 10
+ },
+ {
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+ ],
+ "type": "interline_equation",
+ "angle": 0,
+ "lines": [
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+ "content": "W _ {t} \\leftarrow c \\operatorname {S o f t m a x} \\left(\\left(1 - \\lambda_ {t}\\right) \\log \\left(W _ {t - 1}\\right) - \\eta_ {t} ^ {\\prime} \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right)\\right) \\tag {21}",
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+ "content": " is the soft thresholding operator, applied element-wise. For each element, this operator is defined as"
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+ "content": " can be learned. Notice that since the shrinkage operator is not differentiable, we can approximate it with its smooth approximation. For example, we can use "
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+ "content": ", these choices of attentional bias and retention gate leads to the update rules:"
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+ "content": " is the soft-thresholding operator with parameter "
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+ "type": "text",
+ "content": " , which can be smoothly as explained in Variant 1.1."
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+ "content": " Memory Stability. Existing work is based on the retention gate choices "
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+ "content": "\\mathrm{D}_t(W, W_{t-1}) = \\|W - W_{t-1}\\|_F^2"
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+ "content": "R(W) = \\|W\\|_2^2"
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+ "type": "text",
+ "content": ". However, one can choose other choices of retention gate. For example, in (FTRL Viewpoint), we can choose "
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+ "type": "text",
+ "content": " norm as the regularizer "
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+ "content": "R(W)"
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+ "content": ". More specifically, for "
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+ "type": "inline_equation",
+ "content": "1 < q \\leq 2"
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+ "content": "\\frac {1}{\\eta_ {t}} R (W) = \\frac {1}{2 \\eta (q - 1)} \\| W \\| _ {q} ^ {2}.",
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+ "content": "A_{t} = \\sum_{i = 1}^{t}\\nabla \\ell (W_{i - 1};\\mathbf{k}_{t},\\mathbf{v}_{t})"
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+ "type": "text",
+ "content": "; see Shalev-Shwartz et al. 2012, Section 2.6. Here, "
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+ "type": "text",
+ "content": " denotes the Hadamard (element-wise) product and "
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+ "type": "text",
+ "content": " is the element-wise absolute value operator. Assuming "
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+ "content": ", this update rule can be recursively written as:"
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+ "content": "Variant 5: Bregman Divergence as Retention Gate.. Another natural choice is to use Bregman divergence as retention gate, leading to a mirror descent-type algorithms. In particular, given a smooth strictly convex function "
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+ "type": "inline_equation",
+ "content": "f(\\cdot): \\mathbb{R} \\mapsto \\mathbb{R}"
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+ "type": "text",
+ "content": ". Based on this choice of function "
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+ "type": "text",
+ "content": ", we define the Bregman divergence"
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+ "content": "D _ {t} (W, W ^ {\\prime}) = F (W) - F \\left(W ^ {\\prime}\\right) - \\langle W ^ {\\prime}, W - W ^ {\\prime} \\rangle",
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+ "type": "inline_equation",
+ "content": "\\widetilde{\\ell}_t(W;\\mathbf{k}_t,\\mathbf{v}_t) = \\langle W - W_{t - 1},\\nabla \\ell (W_{t - 1};\\mathbf{k}_t,\\mathbf{v}_t)\\rangle"
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+ "content": " in (Learning-Retaining Viewpoint), we obtain the update rule"
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+ "type": "interline_equation",
+ "content": "W _ {t} = g \\left(- \\eta \\nabla \\ell \\left(W _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) + F ^ {\\prime} \\left(W _ {t - 1}\\right)\\right).",
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+ "content": "F'"
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+ "type": "text",
+ "content": " is the mapping obtained by applying "
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+ "type": "text",
+ "content": " (the derivative of "
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+ "type": "text",
+ "content": ") element-wise to all entries of its input matrix argument. The function "
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+ "type": "inline_equation",
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+ "type": "text",
+ "content": " is the inverse of the mapping "
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+ "type": "text",
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+ "content": "If we choose "
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+ "content": "f(\\tau) = \\frac{\\tau^2}{2}"
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+ "type": "text",
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+ {
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+ "type": "text",
+ "content": " becomes the identity mapping and so is "
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+ "content": "g"
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+ {
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+ "type": "text",
+ "content": ". Therefore, the above update becomes simple gradient descent with no nonlinearity involved in the update rule. However, other choices of "
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+ "content": "f(\\cdot)"
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+ "type": "text",
+ "content": " introduces additional nonlinearity in "
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+ "content": "g(\\cdot)"
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+ "type": "text",
+ "content": ", which can enhance the expressivity of our memory. For example, we can choose the function "
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+ "type": "inline_equation",
+ "content": "f(\\cdot)"
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+ "type": "text",
+ "content": " so that its derivative becomes the inverse sigmoid function, i.e., "
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+ "type": "inline_equation",
+ "content": "f'(\\tau) = \\ln \\left( \\frac{\\tau}{1 - \\tau} \\right)"
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+ "type": "text",
+ "content": " with "
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+ "content": "f': (0,1) \\mapsto \\mathbb{R}"
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+ "type": "text",
+ "content": ". Since "
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+ "type": "inline_equation",
+ "content": "f'(\\cdot)"
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+ "type": "text",
+ "content": " is strictly increasing, then the function "
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+ "type": "text",
+ "content": " (and hence "
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+ "type": "text",
+ "content": ") is strictly convex. Therefore, the Bregman divergence is well defined. Moreover, the inverse of the function "
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+ "content": "f'(\\cdot)"
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+ "type": "text",
+ "content": " becomes the sigmoid function, i.e., "
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+ "type": "inline_equation",
+ "content": "g(\\tau) = \\sigma(\\tau) = \\frac{\\exp(\\tau)}{1 + \\exp(\\tau)}"
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+ "type": "text",
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+ "content": "Architectural Backbone. For the architectural backbone, we fully follow recent studies (Behrouz et al. 2024c; Yang et al. 2024a): We replace attention modules with our variants of MIRAs in Llama's macro architecture with MLPs with SwiGLU(. ) activation, rotary positional encodings (RoPE) (Su et al. 2024), and RMSNorm (Zhang et al. 2019). For MIRAs layer block, we follow the recent modern linear recurrent models (Behrouz et al. 2024c; Yang et al. 2024a), and incorporate a 1D depthwise-separable convolution layer (with kernel size of 4) after each of the query, key, and value projections. For the sake of training stability, we also use "
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+ "type": "text",
+ "content": ". The output of MIRAs layer block is normalized and gated with a linear layer (Mehta et al. 2023)."
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+ "content": "Channel-wise Parameters. For learnable parameters of "
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+ "content": "\\eta_t, \\delta_t"
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+ "type": "text",
+ "content": ". While gaining more expressive power, this parametrization results in significant parameter increase. To mitigate this issue, following Peng et al. (2025b), we use low-rank projections to project the input into "
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+ "content": ", where "
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+ "content": "k"
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+ "type": "text",
+ "content": " is a hyperparameter (usually 32 or 64). The backbone architecture is illustrated in Figure 2."
+ }
+ ]
+ }
+ ],
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+ "content": "Hybrid Models. We also evaluate the hybrid version of Miras's variants. For hybrid models, we follow the Samba (Ren et al. 2024) architecture, in which we sequentially combine our Miras layer with Sliding Window Attention (SWA). The illustration of hybrid model Figure 2."
+ }
+ ]
+ }
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+ "content": "Parallelizable Training. While the design of Miras's variant are theoretically well-motivated, their recurrence is non-linear, potentially making their straightforward training slow for large scales. In this section, we build upon the work of Behrouz et al. (2024c) and Sun et al. (2024) to make the training parallelizable. The main idea is to divide the sequence into"
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+ "content": " (usually is 16 or 64) and calculate the gradient for all tokens in the current chunk with respect to the last state of the memory in the previous chunk. That is, we use "
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+ "type": "inline_equation",
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+ "content": "t'"
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+ "content": " is the last state in the previous chunk."
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+ "type": "text",
+ "content": "Given the above trick, we can calculate all gradients at once and make the recurrence inside each chunk linear. However, to fully take advantage of accelerators, we need to reformulate the process as matrix multiplication. For MONETA, for the sake of clarity, assume "
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+ "type": "inline_equation",
+ "content": "q = 2"
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+ "type": "text",
+ "content": ". We follow the same algorithm as Behrouz et al. (2024c) and expand the recurrence as follows:"
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+ "content": "\\begin{array}{l} \\mathcal {M} _ {t} = \\alpha_ {t} \\mathcal {M} _ {t - 1} - \\eta_ {t} \\nabla \\ell (\\mathcal {M} _ {t - 1}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}) \\\\ = \\beta_ {t} \\mathcal {M} _ {0} - \\sum_ {i = 1} ^ {t} \\eta_ {i} \\frac {\\beta_ {t}}{\\beta_ {i}} \\nabla \\ell \\left(\\mathcal {M} _ {t ^ {\\prime}}; \\mathbf {k} _ {i}, \\mathbf {v} _ {i}\\right), \\tag {28} \\\\ \\end{array}",
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+ "content": "where "
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+ "type": "inline_equation",
+ "content": "t' = t - \\mathrm{mod}(t, b)"
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+ "content": ", and "
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+ "type": "inline_equation",
+ "content": "\\beta_{i} = \\prod_{j=1}^{i} \\alpha_{j}"
+ },
+ {
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+ "type": "text",
+ "content": ". For the sake of clarity, we focus on the first chunk, i.e., "
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+ "type": "inline_equation",
+ "content": "t = b"
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+ "content": " and so "
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+ "content": "t' = 0"
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+ "type": "text",
+ "content": ", and explain the process for the case that "
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+ "type": "inline_equation",
+ "content": "\\mathcal{M}_t = W_t"
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+ "type": "text",
+ "content": " is linear. The process for 2-layer MLPs and other chunks is similar. Using "
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+ "type": "inline_equation",
+ "content": "\\ell_p"
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+ "type": "text",
+ "content": " loss function, we have:"
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+ "type": "interline_equation",
+ "content": "\\begin{array}{l} \\nabla \\ell \\left(W _ {0}; \\mathbf {k} _ {t}, \\mathbf {v} _ {t}\\right) = p \\left(\\operatorname {S i g n} \\left(W _ {0} \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\odot \\left| W _ {0} \\mathbf {k} _ {t} - \\mathbf {v} _ {t} \\right| ^ {p - 1}\\right) \\mathbf {k} _ {t} ^ {\\top} \\\\ \\Rightarrow \\sum_ {i = 1} ^ {b} \\eta_ {i} \\frac {\\beta_ {b}}{\\beta_ {i}} \\nabla \\ell \\left(W _ {0};; \\mathbf {k} _ {i}, \\mathbf {v} _ {i}\\right) = p \\mathbf {E} _ {b} \\odot \\mathbf {B} _ {b} \\odot \\operatorname {S i g n} \\left(W \\mathbf {k} _ {t} - \\mathbf {v} _ {t}\\right) \\odot \\left(\\left| W _ {0} \\mathbf {K} - \\mathbf {V} \\right| ^ {p - 1}\\right) \\mathbf {K} ^ {\\top}, \\tag {29} \\\\ \\end{array}",
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+ }
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+ "content": "where "
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+ "type": "inline_equation",
+ "content": "\\mathbf{E}_b = \\left[\\eta_1\\quad \\eta_2\\quad \\dots \\quad \\eta_b\\right]"
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+ "content": " and "
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+ "type": "inline_equation",
+ "content": "\\mathbf{B}_b"
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+ "type": "text",
+ "content": " is defined analogously on "
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+ "type": "inline_equation",
+ "content": "\\frac{\\beta_b}{\\beta_i}\\mathrm{s}"
+ },
+ {
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+ "type": "text",
+ "content": ". For the sake of stability in training, we use "
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+ "type": "inline_equation",
+ "content": "\\operatorname{Sign}(x)\\approx \\tanh (\\alpha x)"
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+ "type": "text",
+ "content": " and "
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+ "type": "inline_equation",
+ "content": "|x| = \\sqrt{x^2 + \\epsilon}"
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+ "type": "text",
+ "content": ", where "
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+ "type": "inline_equation",
+ "content": "\\epsilon >0"
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+ {
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+ "type": "text",
+ "content": " is a small number (i.e., "
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+ {
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+ "type": "inline_equation",
+ "content": "\\epsilon = 1e - 6"
+ },
+ {
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+ ],
+ "type": "text",
+ "content": "). As discussed in Equation 24, the case that "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "q\\neq 2"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " appears as a normalization term on the memory. Similar to Titans (Behrouz et al. 2024c) and TTT (Sun et al. 2024), we do not apply this non-linearity inside each chunk and instead use it at the end of each chunk."
+ }
+ ]
+ }
+ ],
+ "index": 5
+ },
+ {
+ "bbox": [
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+ "type": "text",
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+ "spans": [
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+ ],
+ "type": "text",
+ "content": "For YAAD, the process is very similar to the above. We calculate the gradient of both "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\ell_1"
+ },
+ {
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+ "type": "text",
+ "content": " and "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\ell_2"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " loss and use a masking based on "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\| \\mathcal{M}(\\mathbf{k}_t) - \\mathbf{v}_t\\| \\leq \\delta_t"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "."
+ }
+ ]
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "angle": 0,
+ "lines": [
+ {
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+ "spans": [
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+ "type": "text",
+ "content": "For MEMORA, the update rule has two non-linear parts, i.e., softmax and log, making the model hardly parallelizable. To this end, as discussed above, we use its linear version inside each chunk and its non-linear version across chunks. However, using both log and softmax at the end of each chunk removes the effect of log. To this end, we consider a lag tokens after each chunk (i.e., tokens with index "
+ },
+ {
+ "bbox": [
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+ "type": "inline_equation",
+ "content": "i = kb + 1"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": ", where "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "b"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": " is the chunk size and "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "k \\in \\mathbb{Z}^+"
+ },
+ {
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+ "type": "text",
+ "content": "). That is, let "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "\\mathcal{M}_0"
+ },
+ {
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+ "content": "Setup. We train our models with training context window of size 4096 using either FineWeb-Edu dataset (Penedo et al. 2024) (for LM and common-sense reasoning tasks) or C4 dataset (Raffel et al. 2020) (for scaling patterns). We use model"
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+ "content": "sizes of 120M, 340M, 760M, and 1.3B parameters. We train small models (120M and 340M) on 15B tokens sampled from the dataset, the medium size model (760M) on 30B tokens, and the large model on 100B tokens. Baseline results are reported by Behrouz et al. (2024c)."
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+ "content": "We follow recent studies (Behrouz et al. 2024c; Yang et al. 2024a,c) and first focus on the perplexity in language modeling and also commonsense reasoning tasks. The results for MEMORA, YAAD, MONETA and also baselines with size of 340M, 760, and 1.3B are reported in Table 2. All of our variants outperforms all the baselines including Transformer++, modern linear recurrent models and hybrid methods. The superior performance compared to hybrid models is particularly important as all of our variants are pure recurrent (attention-free). Among the three variants of MirAS, while MONETA achieves slightly weaker performance than MEMORA, and YAAD, the other two variants are close and depending on the task and model size, the best model can vary."
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+ "content": "Context Length. We first vary the training context length from 2K to 32K for two versions of our model with size 340M and 760M. The results are reported in Figure 3 (Middle and Right). All three variants of Miras scales better than state-of-the-art baselines when increasing the context length. We attribute this superior performance to: (1) expressive memory architecture. Contrary to baselines like Mamba2 and GSA that uses vector- and matrix-valued memory, our variants are using 2-layer MLPs with more expressive power to learn from longer sequences. (2) The choice of retention gate and attentional bias: All of our three variants go beyond the standard attentional biases and retention gates. These choices can help the memory to better manage its fixed-size capacity."
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+ "content": "Model Size. We also report the #FLOPs vs. perplexity of our models and baselines in Figure 3 (Left). All three variants outperforms all baselines given almost the same budget of FLOPs. These results, once again support the importance of powerful memory design."
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+ "html": "| Task | Prompt |
| FGQA | Question: [question] \\n Options: \\n (A) [option1] \\n (B) [option2] \\n Only give the best option. |
| SGQA | The following question is asked by the camera wearer at the end of the video. Provide a detailed answer even if unsure. Try to answer in around 20-30 words. Now answer the following question based on the video content: [question] |
| RDCap | Create a dense caption of the subject's actions within the red rectangles, including action frames ids and brief descriptions. For each item use the format [start, end]: [description] separated by a newline, where start and end are frame numbers between 0 and 31 in this 32 frame video. |
| RCap | Give a detailed description of the events occurring in the region marked by the red rectangle within frames ([start frame], [end frame]) in this 32 frame video |
| RTLoc | Given the region marked by the red rectangle in the video, please provide the start and end frame of when '[event]' happens. Use the format (start, end), where start and end are frame numbers between 0 and 31 in this 32 frame video. |
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+ "text": "D Additional PLM-VideoBench Results",
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+ "text": "We present benchmarking results across all model scales (1B, 3B, 8B) in Table 11, to supplement the 8B model results in the main paper (Table 5). Our approach consistently outperforms baselines across all scales, including proprietary models whose model scale is unknown.",
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+ "Table 10: PLM-VideoBench task prompts. Items in square brackets are placeholders filled in for each benchmark instance."
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+ "table_footnote": [],
+ "table_body": "| Question Type | Sample Questions |
| Action Recognition | What is the process being performed on the sandpaper? \nWhat is the action shown? |
| Action Sequence | What does the person do after brewing the tea? \nWhat does the person do before marking the vinyl with a pencil? |
| Counting Problems | What is the quantity of universal down cleaner being poured into the task area? \nHow many branches does the person cut in total? \nHow many times does the person spray Greased Lightning onto the ketchup spill? |
| Movement Direction | In what direction is the black welding tool pointing while the person is working on the metal joint? \nHow does the person chop the garlic with the knife? |
| Object Attributes | What is the color of the seatpost shown in the video segment? \nWhat is the shape of the tube at the end of the step? \nWhat is the size of the knife being used to chop the spring onions? |
| Object Location | Where does the person put the honey bottle away? \nWhere does the person position the clothes before ironing? |
| Object Recognition | What type of roller and paint are being used? \nWhat does the person place on top of the smooth half of the egg carton? \nWhat was the person initially holding in their left hand? |
| Object State | How would you describe the sink at the beginning of the cleaning process? \nWhat is the state of the nematode after mixing it with water and sponge? |
| Other | At what point in the video is the person seen holding the wires? |
| Pose | How are the woman's legs positioned while she is sitting? \nHow bent is the left elbow during the activity? |
| Spatial Relations | How far is the bias tape maker from the right edge of the ironing board? \nWhat is the spatial relationship between the bowls and the Brussels sprouts on the kitchen countertop? |
| Speed/Force | How would you describe the consistency of pressure applied during sanding? \nHow fast does the person initially push the stone? |
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+ "text": "for scene classification information. We incorporate the textual annotations directly into language prompts using the following template:",
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| FGQA | Question: [question] \\n Options: \\n (A) [option1] \\n (B) [option2] \\n Only give the best option. |
| SGQA | The following question is asked by the camera wearer at the end of the video. Provide a detailed answer even if unsure. Try to answer in around 20-30 words. Now answer the following question based on the video content: [question] |
| RDCap | Create a dense caption of the subject's actions within the red rectangles, including action frames ids and brief descriptions. For each item use the format [start, end]: [description] separated by a newline, where start and end are frame numbers between 0 and 31 in this 32 frame video. |
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| RTLoc | Given the region marked by the red rectangle in the video, please provide the start and end frame of when '[event]' happens. Use the format (start, end), where start and end are frame numbers between 0 and 31 in this 32 frame video. |
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| Action Recognition | What is the process being performed on the sandpaper? \nWhat is the action shown? |
| Action Sequence | What does the person do after brewing the tea? \nWhat does the person do before marking the vinyl with a pencil? |
| Counting Problems | What is the quantity of universal down cleaner being poured into the task area? \nHow many branches does the person cut in total? \nHow many times does the person spray Greased Lightning onto the ketchup spill? |
| Movement Direction | In what direction is the black welding tool pointing while the person is working on the metal joint? \nHow does the person chop the garlic with the knife? |
| Object Attributes | What is the color of the seatpost shown in the video segment? \nWhat is the shape of the tube at the end of the step? \nWhat is the size of the knife being used to chop the spring onions? |
| Object Location | Where does the person put the honey bottle away? \nWhere does the person position the clothes before ironing? |
| Object Recognition | What type of roller and paint are being used? \nWhat does the person place on top of the smooth half of the egg carton? \nWhat was the person initially holding in their left hand? |
| Object State | How would you describe the sink at the beginning of the cleaning process? \nWhat is the state of the nematode after mixing it with water and sponge? |
| Other | At what point in the video is the person seen holding the wires? |
| Pose | How are the woman's legs positioned while she is sitting? \nHow bent is the left elbow during the activity? |
| Spatial Relations | How far is the bias tape maker from the right edge of the ironing board? \nWhat is the spatial relationship between the bowls and the Brussels sprouts on the kitchen countertop? |
| Speed/Force | How would you describe the consistency of pressure applied during sanding? \nHow fast does the person initially push the stone? |
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| SGQA | The following question is asked by the camera wearer at the end of the video. Provide a detailed answer even if unsure. Try to answer in around 20-30 words. Now answer the following question based on the video content: [question] |
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| RTLoc | Given the region marked by the red rectangle in the video, please provide the start and end frame of when '[event]' happens. Use the format (start, end), where start and end are frame numbers between 0 and 31 in this 32 frame video. |
+
+# D Additional PLM-VideoBench Results
+
+We present benchmarking results across all model scales (1B, 3B, 8B) in Table 11, to supplement the 8B model results in the main paper (Table 5). Our approach consistently outperforms baselines across all scales, including proprietary models whose model scale is unknown.
+
+Table 10: PLM-VideoBench task prompts. Items in square brackets are placeholders filled in for each benchmark instance.
+
+| Question Type | Sample Questions |
| Action Recognition | What is the process being performed on the sandpaper?
+What is the action shown? |
| Action Sequence | What does the person do after brewing the tea?
+What does the person do before marking the vinyl with a pencil? |
| Counting Problems | What is the quantity of universal down cleaner being poured into the task area?
+How many branches does the person cut in total?
+How many times does the person spray Greased Lightning onto the ketchup spill? |
| Movement Direction | In what direction is the black welding tool pointing while the person is working on the metal joint?
+How does the person chop the garlic with the knife? |
| Object Attributes | What is the color of the seatpost shown in the video segment?
+What is the shape of the tube at the end of the step?
+What is the size of the knife being used to chop the spring onions? |
| Object Location | Where does the person put the honey bottle away?
+Where does the person position the clothes before ironing? |
| Object Recognition | What type of roller and paint are being used?
+What does the person place on top of the smooth half of the egg carton?
+What was the person initially holding in their left hand? |
| Object State | How would you describe the sink at the beginning of the cleaning process?
+What is the state of the nematode after mixing it with water and sponge? |
| Other | At what point in the video is the person seen holding the wires? |
| Pose | How are the woman's legs positioned while she is sitting?
+How bent is the left elbow during the activity? |
| Spatial Relations | How far is the bias tape maker from the right edge of the ironing board?
+What is the spatial relationship between the bowls and the Brussels sprouts on the kitchen countertop? |
| Speed/Force | How would you describe the consistency of pressure applied during sanding?
+How fast does the person initially push the stone? |
+
+Table 17: PLM-FGQA question types and sample questions
+
+for scene classification information. We incorporate the textual annotations directly into language prompts using the following template:
+
+# Automatic answer generation prompt
+
+A video is showing a task [video level task name], specifically the part where [ASR caption]. Here is what we already know about the video: [existing question-answer pairs]. Answer this question in detail: [question]
+
+The off-the-shelf embeddings are incorporated into the PLM input via an additional Perceiver-IO[227] tokenizer, which summarizes the embeddings at the segment level.
+
+We fine-tune the answer generator on 1M manually annotated QA pairs. After fine-tuning, we deploy the trained answer generator with privileged information access on the unlabelled questions produced in the previous step, to produce automatic answers.
+
+# G.1.4 Human Annotation
+
+After obtaining segments and generating questions and automatic answers, we employ human annotators to obtain high-quality answers. Our answer annotations include the following:
+
+- Human-verified answers: Raters are provided with the model-generated answer and are asked to accept or reject the answer. They can reject questions for being irrelevant or unanswerable, and answers for being factually incorrect or lacking details. Accepted question-answer pairs proceed without changes, while rejected ones are handled differently:
+
+question-related rejections (irrelevant or unanswerable) are discarded, whereas answer-related rejections (factually incorrect or lacking details) are marked for correction in the next phase. $17.8\%$ of the total training samples are human-verified automatic answers.
+
+- Human annotated answers: Raters answer the questions from scratch by ensuring to cover all the relevant details within the temporal segment. They receive reference information, such as video-level task names and ASR captions, and may use online resources like WikiHow for additional context. Questions that cannot be answered based on the video segment (for example, due to some false premise) are rejected (with an explanation). These manually annotated answers make up $82.2\%$ of the PLM-FGQA training split, and $100\%$ of the evaluation set.
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+Quality Control. Data quality is crucial for model success. We followed several strategies to monitor and enhance annotation quality: annotation Certification - we reviewed a small sample of annotations from each rater before they could work in production queues, ensuring that annotators met high-quality standards before advancing to production; golden Examples - annotators were provided with high-quality annotation examples, highlighting common error patterns and offering acceptable answers. targeted and Dual QA - we conducted daily audits, including vendor auditing and our own sampled quality control. In total, $13\%$ of the training set was audited, and $100\%$ of the samples in PLM-VideoBench underwent quality control.
+
+# G.2 FGQA PLM-VideoBench Construction
+
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| RTLoc | Given the region marked by the red rectangle in the video, please provide the start and end frame of when '[event]' happens. Use the format (start, end), where start and end are frame numbers between 0 and 31 in this 32 frame video. |
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+ "html": "| Algorithm Category | Reference |
| Specialized Methods | § 5Feature-aspect Methods | Feature Encoding | [33], [45], [64] |
| Feature Selection | [59], [60], [105], [61], [193] |
| Feature Projection | [52], [33], [34], [58] |
| Feature Interaction | [54], [62], [63], [55], [65], [49], [215] |
| § 4.1Sample-aspect Methods | Sample Interaction | [70], [216], [217], [192], [67] |
| Neighbor Retrieval | [218], [68], [69], [35] |
| § 4.2Objective-aspect Methods | Training Objective | [67] |
| Training Regularization | [219], [50], [66] |
| § 6Transferable Methods | Homogeneous | [63], [48], [70], [220], [46], [221], [222], [223], [47], [224], [225], [226], [227] |
| Heterogeneous | [228], [229], [222], [72], [73], [64], [230], [231] |
| Language Model | [77], [232], [182], [79], [78], [233], [234], [82], [83], [235], [236], [80], [237] |
| Vision Model | [238], [239], [240], [74], [75], [241], [242], [76] |
| § 7General Mehtods | Raw-Feature-based | [86], [87], [88] |
| TabPFN Variants | [89], [91] |
| Semantics-based | [92], [93], [94], [243] |
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+ "text": "where $\\mathcal{D}$ is the set of all samples (training data) available for retrieval or learning. $\\mathcal{R}(\\cdot)$ is the sample interaction module, which retrieves or aggregates information from relevant samples in $S$ for the target sample $\\boldsymbol{x}_i$ . $\\Phi$ represents the learnable parameters of $\\mathcal{R}$ . $f(\\cdot)$ is the prediction head that maps the aggregated information to the final output $\\hat{y}_i$ .",
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+ "text": "extends this via non-parametric Transformers, whereas Hopular [217] employs Hopfield networks, sharing conceptual alignment with SAINT [70]. Unlike nearest-neighbor classification, the distance metric is learned end-to-end. Prompt [192] posits that the feature importance in tabular data is sample-dependent. During feature extraction, it treats the information between samples as prompts. PTaRL [67] identifies two issues in the representation of tabular data samples: entanglement and localization. It addresses these by modeling global sample relationships through prototype generation and representation projection, helping the model produce clear and consistent decisions.",
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+ "text": "DNNR [68] argues that a key advantage of neighbor-based methods is the model's transparency, meaning that the model's decisions can be explained by inspecting its components. It enhances predictive performance by incorporating local gradient estimation and Taylor series approximation into the KNN framework. TabR [69] proposes that, compared to purely parametric (e.g., retrieval-free) models, retrieval-based models can achieve superior performance while also exhibiting several practically important properties, such as the ability for incremental learning and enhanced robustness. It encodes all candidate samples and then employs an attention-like mechanism to retrieve the samples that aid in making predictions, as explored in [218]. ModernNCA [35] revitalizes the classic tabular prediction method, Neighbourhood Component Analysis (NCA) [245], by designing and incorporating deep learning architectures and strategies. The resulting method efficiently leverages neighboring samples for prediction.",
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+ "text": "bles the current in-context learning [246] mechanism. In particular, the in-context learning used in general models like TabPFN [89], [91] can aslo be considered a form of the neighborhood method. This concept of neighborhood not only helps in standard tasks, but also enhances transferable and general methods. For example, LoCalPFN [90] highlights that employing local linear regression can lead to more expressive decision boundaries, while utilizing local context allows performance to scale with the size of the training dataset.",
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+ "text": "(Ridge) [248], or Elastic-Nets [249] penalize large coefficients, effectively controlling the complexity of the model and helping to maintain a balance between bias and variance.",
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+ "text": "Objective-aspect methods in deep learning are an extension of these traditional regularization techniques, where inductive bias is introduced by adjusting the loss function $\\ell$ or adding regularizers $\\Omega$ . In the training process, the goal is to leverage regularization on the model to improve predictions.",
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+ "text": "Training Regularization. For training regularization, RLNs [219] overcome the challenge of an intractable number of hyperparameters during training by introducing an efficient tuning scheme, which minimizes a new \"Counterfactual Loss.\" In RLNs, the regularization coefficients are optimized together with learning the network weight parameters. RLNs produce extremely sparse networks, thus providing more interpretable models and revealing the importance that the network assigns to different inputs. [50] introduces \"cocktails,\" dataset-specific combinations of 13 regularization techniques, showing that even simple neural networks can outperform tree-based architectures when optimized with these methods. TANGOS [66] introduces a regularization-based improvement. It regularizes neuron attributions to encourage neurons to specialize and become orthogonal to one another.",
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+ "text": "5 FEATURE-ASPECT SPECIALIZED METHODS",
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+ "text": "Tabular data is characterized by a diverse set of features, including both categorical and numerical variables. The complexity of tabular data arises from the variety of feature types, their interrelationships, and the high dimensionality often present. Traditional methods often rely on manual feature engineering, using techniques such as encoding categorical variables and selecting relevant features to improve model performance and reduce overfitting.",
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+ "text": "As deep learning has evolved, these traditional techniques have been integrated and expanded upon. Deep tabular models are capable of automatically learning complex feature representations, reducing the need for explicit feature engineering. Feature-aspect methods, such as feature encoding, selection, projection, and interaction, are essential for transforming raw tabular inputs into more informative intermediate forms. These methods help improve a model's ability to capture intricate relationships between features, thereby enhancing its generalization capabilities.",
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+ "text": "Categorical Encoding. Categorical variables represent types of data which may be divided into groups. Examples of categorical variables are race, sex, age group, and educational level [250]. The categorical features are usually transformed in an index (integer). The two most popular techniques are an Ordinal Encoding and a One-Hot Encoding.",
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+ "text": "Ordinal Encoding assigns each unique category a distinct integer value. This approach is useful when the categorical variable has an inherent order, such as \"low,\" \"medium,\" and \"high.\" The main advantage of Ordinal Encoding is its simplicity and efficiency, as it transforms the categorical variable into a single numeric column. However, it assumes that there is an ordinal relationship between the categories, which may not always be the case. For instance, if the categorical variable represents \"color\" with categories such as \"red,\" \"blue,\" and \"green,\" applying Ordinal Encoding would introduce an artificial order that does not reflect any meaningful ranking.",
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+ "text": "On the other hand, One-Hot Encoding creates a new binary column for each unique category in the original categorical variable. For example, for a variable \"color\" with three categories (red, blue, and green), One-Hot Encoding would generate three binary columns: \"is_red,\" \"is_green,\" and \"is_green,\" encoding red as $(1,0,0)$ , blue as $(0,1,0)$ and green as $(0,0,1)$ . Each column indicates the presence or absence of that particular category. One-Hot Encoding is useful for nominal categorical variables, where no order exists between the categories. While One-Hot Encoding avoids the assumption of ordinal relationships, it can lead to a high-dimensional feature space if the categorical variable has many unique values, which may result in increased computational costs and potential issues with overfitting.",
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+ "type": "text",
+ "text": "In some cases, more advanced encoding techniques are used to address the limitations of these basic approaches.",
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+ "text": "For example, Target Encoding assigns each category a value based on the mean of the target variable for that category. This method can be useful when there is a strong relationship between the categorical features and the target. In Leave-one-out embedding, every category is replaced with the mean of the target variable of that category, which excludes the current row to avoid overfitting.",
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+ "text": "Numerical Encoding. For encoding, MLP-PLR [45] introduces two numerical encoding methods: Piecewise Linear Encoding (PLE) and Periodic Activation Functions. These encoding methods can be integrated with other differentiable layers (e.g., Linear, ReLU) to enhance performance. PLE produces alternative initial representations for the original scalar values and is based on feature binning. Periodic Activation Functions take into account the fact the embedding framework where all features are computed independently of each other forbids mixing features during the embedding process and train the pre-activation coefficients instead of keeping them fixed. [38] utilizes tools from spectral analysis, showing that functions described by tabular datasets often have high irregularity, and can be smoothed by transformations such as scaling and ranking to improve performance. They propose \"frequency reduction\" as an inductive bias during training.",
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+ "type": "text",
+ "text": "Feature Tokenization. Feature tokenizer performs a similar role to the feature extractor in traditional models. It transforms the input features to embeddings [62], [33]. Since the feature representations of features are very sparse and high-dimensional, a common way is to represent them into low-dimensional spaces (e.g., word embeddings).",
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+ "text": "The general form for feature tokenization can be expressed as:",
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+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {T} _ {i, j} = \\boldsymbol {b} _ {j} + \\mathcal {T} \\left(x _ {i, j}; \\Psi\\right) \\in \\mathbb {R} ^ {t}, \\tag {3}\n$$\n",
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+ "type": "text",
+ "text": "where $\\mathcal{T}(\\cdot)$ is the feature tokenizer module, which transforms the input feature vector $\\pmb{x}_i\\in \\mathbb{R}^d$ to a token embedding $T_{i,j}\\in \\mathbb{R}^t$ . $t$ is the dimension of token embedding. $\\pmb{b}_{j}$ is the $j$ -th feature bias. $\\mathcal{T}$ can be implemented with different forms. $\\Psi$ represents the learnable parameters of $\\mathcal{T}$",
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+ "type": "text",
+ "text": "In AutoInt [62], both the categorical and numerical features are embedded into low-dimensional spaces, which reduces the dimension of the input features and meanwhile allows different types of features to interact with each other. The embeddings of categorical features are computed by multiplying the embedding matrix with the multi-hot vector, while a corresponding embedding vector represents numerical features. TabTransformer [63] embed each categorical feature into a parametric embedding of dimension $t$ using Column embedding. An embedding vector is assigned to each feature, and a set of embeddings is constructed for all categorical features. Unlike TabTransformer, SAINT [70] proposes projecting numerical features into a $t$ -dimensional space before passing their embedding through the transformer encoder. FT-Transformer [33] adapts the Transformer architecture for tabular data, where all features are transformed to embeddings and applies a stack of Transformer layers to the embeddings. Specifically, the numerical tokenizer is implemented as the element-wise multiplication $\\boldsymbol{T}_i^{\\mathrm{num}} = \\boldsymbol{b}_i^{\\mathrm{num}} + x_i^{\\mathrm{num}} \\cdot \\boldsymbol{W}_i^{\\mathrm{num}}$ , and the categorical tokenizer is implemented as the lookup table $\\boldsymbol{T}_i^{\\mathrm{cat}} = \\boldsymbol{b}_i^{\\mathrm{cat}} + \\boldsymbol{e}_i^T \\boldsymbol{W}_i^{\\mathrm{cat}}$ , where $\\boldsymbol{e}_i^T$ is a one-hot vector for the corresponding categorical feature. Other transformer-based",
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+ {
+ "type": "header",
+ "text": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX",
+ "bbox": [
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+ "page_idx": 10
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+ "type": "page_number",
+ "text": "11",
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+ "type": "text",
+ "text": "methods, like [65], [72], [230], [215], use the same feature tokenizer as FT-Transformer.",
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+ "type": "text",
+ "text": "5.2 Feature Selection",
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+ "type": "text",
+ "text": "The high dimensionality of tabular data often causes overfitting, where the model focuses on irrelevant features and neglects the important ones. Feature selection reduces the number of features, retaining only the most valuable information. This helps prevent overfitting, improves generalization, and reduces computational complexity.",
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+ "type": "text",
+ "text": "Traditional tree-based models facilitate automatic feature selection by evaluating the impact of each feature on the target during the construction process. Decision trees utilize metrics such as information gain or the Gini index for feature selection, while ensemble methods like random forests determine feature importance by assessing each feature's contribution [251], [252], [253]. Recently, modern deep learning methods for tabular data often mimic trees' structures for feature selection.",
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+ "type": "text",
+ "text": "GrowNet [59] and NODE [60] primarily mimic ensemble techniques. Inspired by GBDT, GrowNet designs a framework for building DNNs with multiple weak learners, where each learner's input consists of the original features plus the penultimate layer output from the previous learner. NODE uses a differentiable Oblivious Decision Tree as the base model, applying Bagging within each layer and Stacking across layers in a multi-layered structure. To make GAM [254] scalable and effective, NODE-GAM [61] modifies NODE to be a GAM, allowing GAM to learn quick, nonlinear jumps that better match patterns in real data.",
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+ "type": "text",
+ "text": "TabNet [105] and GRANDE [193] focus more on how tree models handle features. TabNet not only retains the representation learning capabilities of DNNs through self-supervised learning, but also incorporates the interpretability of tree models and the benefits of sparse feature selection, with a model structure designed for both feature selection and computation. GRANDE argues that the hard splits used by tree models are a key advantage over deep models, and thus proposes a method for learning hard, axis-aligned tree ensembles using gradient descent. GRANDE combines the beneficial inductive bias of axis-aligned splits with the flexibility provided by gradient descent optimization.",
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+ "type": "text",
+ "text": "5.3 Feature Projection",
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+ "type": "text",
+ "text": "Feature projection methods aim to project the raw data into a middle form, enhancing the representation ability for later architectures. Feature projection methods can be broadly categorized into two main approaches: MLP variants and special designed architectures. These approaches aim to enhance the model's ability to represent complex features for underlying feature structures.",
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+ "type": "text",
+ "text": "MLP Variants. For model architecture, RTDL [33] investigates both ResNet-like and Transformer-based architectures tailored for tabular data, proposing simple yet effective adaptations of these widely-used deep models. In particular, the MLP architecture is constructed by stacking multiple blocks consisting of Linear layers, ReLU activations, and Dropout, which transform the raw tabular features into a fixed-dimensional hidden representation. A final linear layer is then used as the classification head. The paper highlights",
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+ {
+ "type": "text",
+ "text": "an important insight: with proper hyperparameter tuning, even simple architectures like MLP and ResNet can achieve competitive performance on tabular benchmarks.",
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+ "type": "text",
+ "text": "Another contemporaneous work [50] enhances the MLP architecture by equipping it with a comprehensive suite of modern regularization techniques. Instead of introducing architectural innovations, this study focuses on systematically exploring combinations of 13 different regularization methods to identify an effective \"regularization cocktail\" for plain MLPs. The results demonstrate two key findings: (i) a well-regularized vanilla MLP can significantly outperform many recent, specialized neural architectures designed for tabular data; and (ii) such MLPs can even surpass strong traditional machine learning models like XGBoost across a range of benchmarks. For a more comprehensive strategy, RealMLP [34] explores multiple aspects including preprocessing, hyperparameters, architecture, regularization, and initialization.",
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+ "type": "text",
+ "text": "Special Designed Architectures. For units, motivated by the observation that normalization techniques are prone to disturbances during training, SNN [52] proposes the Scaled Exponential Linear Unit (SELU) to improve deep models for tabular data. NAMs [255] uses exp-centered (ExU) hidden units to improve the learnability for fitting jumpy functions. BiSHop [58] uses a dual-component approach, sequentially processing data both column-wise and row-wise through two interconnected directional learning modules. They use layers of generalized sparse modern Hopfield layers, a sparse extension of the modern Hopfield model with learnable sparsity.",
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+ "type": "text",
+ "text": "5.4 Feature Interaction",
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+ "text": "Feature interaction methods aim to model relationships among features to enhance the representation power of deep learning models on tabular data. In tabular datasets, each sample $\\boldsymbol{x}_i \\in \\mathbb{R}^d$ is described by $d$ features, and the goal is to transform these raw features into enriched representations that improve predictive performance.",
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+ "type": "text",
+ "text": "The general form for feature interaction methods can be expressed as:",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\hat {y} _ {i} = f \\left(\\mathcal {H} \\left(\\boldsymbol {x} _ {i}; \\Theta\\right)\\right), \\tag {4}\n$$\n",
+ "text_format": "latex",
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+ {
+ "type": "text",
+ "text": "where $\\pmb{x}_i\\in \\mathbb{R}^d$ is the input feature vector for a single instance, $\\mathcal{H}(\\cdot)$ is the feature interaction module, which transforms the input $\\pmb{x}$ by capturing feature dependencies or generating higher-order feature interactions. $\\Theta$ represents the learnable parameters of $\\mathcal{H}$ . $f(\\cdot)$ is the prediction head that maps the transformed representation to the final output $\\hat{y}$ .",
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+ "type": "text",
+ "text": "Feature interaction methods can be broadly categorized into two main approaches: the design of automatic feature interaction modules and the mining of implicit feature relationships. These approaches aim to enhance the model's ability to learn complex feature interactions and underlying feature structures within tabular data.",
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+ "type": "text",
+ "text": "Automatic Feature Interaction Modules. These methods do not assume specific feature types within the tabular dataset. Instead, they focus on improving the feature interaction process, enabling the model to learn complex, high-order feature relationships autonomously.",
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+ "type": "text",
+ "text": "DCNv2 [54] improves the learning of the model's feature interaction by improving the \"Cross Network\" structure. It",
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+ {
+ "type": "header",
+ "text": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "page_number",
+ "text": "12",
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+ {
+ "type": "text",
+ "text": "employs low-rank methods to approximate feature crosses in subspaces and then integrates these subspaces using a gating mechanism. AutoInt [62] maps the original sparse high-dimensional feature vectors into a low-dimensional space and models high-order feature interactions by stacking interaction layers with a multi-head attention mechanism. Unlike AutoInt, the TabTransformer[63] only maps categorical features into contextual embeddings and feeds them into a Transformer model, while numerical continuous features are directly concatenated with the interacted contextual embeddings. When tabular data contains only numerical features, TabTransformer behaves in an MLP-like manner. Conversely, when the data contains only categorical features, TabTransformer operates similarly to AutoInt.",
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+ "type": "text",
+ "text": "Implicit Feature Relationships. Methods in this category typically assume that features in tabular data can be abstracted into implicit types and that it is necessary to design a suitable feature learning process to adapt to the characteristics of different types of features.",
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+ "type": "text",
+ "text": "DANets [55] propose the existence of underlying feature groups in tabular data, where features within each group are correlated. They learn to group input features and perform further feature abstraction. SwitchTab [49] introduces the idea of extracting sample-specific \"Salient Features\" and sample-shared \"Mutual Information\" in tabular features. It leverages self-supervised learning to assist in learning feature representations. ExcelFormer [65] argues that while DNN assigns weights to each feature, it does not actively exclude irrelevant features. To address this, it introduces Semi-Permeable Attention for feature interaction, which allows features with lower information content to access information from more informative features while preventing highly informative features from being influenced by less relevant ones. AMFormer [215] proposes the hypothesis that arithmetic feature interactions are crucial for deep tabular models. Based on the Transformer architecture, it introduces components designed to extract both additive and multiplicative interaction information.",
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+ {
+ "type": "text",
+ "text": "6 FROM SPECIALIZED TO TRANSFERABLE MODEL",
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+ "type": "text",
+ "text": "Instead of training a tabular model from scratch, learning based on a Pre-Trained Model (PTM) may increase the learning efficacy and reduce the resource and data requirement. For example, in a house prices prediction task, training a regressor in a certain area may benefit from a well-trained predictor from its neighborhood.",
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+ {
+ "type": "text",
+ "text": "Learning by reusing the PTM usually contains two stages. The first is the pre-training of a tabular model, from one or more upstream tasks. Given the PTM and a downstream task, an adaptation strategy is needed to transform the PTM to the target task or facilitate the learning of the target model. Formally, a well-trained model $g_{\\Theta}$ is often available and can be leveraged to facilitate the training of $f_{\\theta}$ over $\\mathcal{D}$ . Here, $g_{\\Theta}$ is pre-trained on a dataset $\\mathcal{D}' = \\{(x_j', y_j')\\}_{j=1}^{N'}$ with instances $x_j' \\in \\mathbb{R}^{d'}$ and labels $y_j' \\in [C']$ . To reuse expert knowledge in $g_{\\Theta}$ , an adaptation strategy is applied: $f_{\\theta} = \\text{Adapt}(f_{\\theta_0} \\mid \\mathcal{D}, g_{\\Theta})$ , where $\\theta_0$ is the initialization of the model. The notation could also be extended to cases with more than one PTM. The main challenge to reuse one or more PTMs is to bridge the gap between the PTM and the",
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+ "type": "text",
+ "text": "target tabular model [256]. We categorize PTMs into three kinds based on the source of PTM $g_{\\Theta}$ .",
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+ "type": "text",
+ "text": "Homogeneous Transferable Tabular Model. First, the PTM may come from the same form of task (with $d' = d$ and $C' = C$ , but with different distributions $\\operatorname{Pr}(\\mathcal{D}') \\neq \\operatorname{Pr}(\\mathcal{D})$ or model families $g \\neq f$ ). For example, those pre-trained from other domains [71], or those unlabeled instances [48], [70].",
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+ "type": "text",
+ "text": "Heterogeneous Transferable Tabular Model. In addition, we consider a PTM pre-trained from a slightly different task with $\\mathcal{D}$ . In addition to the previous difference, the PTM $g_{\\Theta}$ may differ from $f_{\\theta}$ in feature dimension $(d' \\neq d)$ or target class set $(C' \\neq C)$ , so the adaptation method $\\mathbf{Adapt}(\\cdot)$ must handle such heterogeneity [64], [230].",
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+ "type": "text",
+ "text": "Cross-Modal Transferable Tabular Model. Moreover, the pre-trained model could also be constructed from another modality, such as vision and language domains. The cross-modality PTM is hard to be applied to the tabular prediction task in most cases, so auxiliary information from the tabular task like the semantic meaning of attributes (i.e., the attribute names) are usually assumed to be available in this case, where PTM like large language models may provide the latent semantic meanings as external knowledge [77], [73].",
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+ "type": "text",
+ "text": "The main limitation of the transferable tabular model is the assumption that the data distribution of the well-trained model should be similar to the distribution of the target model. For example in the previous house price prediction task, if the PTM is pre-trained in an area distance from the target area and targets diverse problems, it is hard to utilize the PTM in the target task [222]. Since different tabular tasks may vary in their distribution, feature, or classes, the general assumption is their exist a common \"dimension\" between the PTM and the target task. Only the distribution changes under the shared dimension and classes, or there exists an overlap between the feature or class spaces [230]. For example, in real-world applications such as healthcare, there are numerous medical diagnostic tables. These tables usually have some features in common such as blood type and blood pressure. For rare diseases with limited data, knowledge transfer from other diagnostic tables with overlapping features becomes beneficial [228]. When the feature/label semantics are available, two different tasks may be linked through the semantic space, and textual PTMs can be used to map the tabular instance to this space or facilitate the prediction in this space [80].",
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+ "type": "text",
+ "text": "Pros and Cons of transferable Models. Learning with a well-trained tabular model has several advantages based on the knowledge encoded in the PTM. First, the training efficiency of the target model is improved and the model may converge fast, as the PTM may provide better initialization weights or optimization paths. Then, the target model will reduce the requirement on the data size, i.e., learning with a few-shot dataset. Training based on a PTM also reduces the number of learnable parameters, leading to parameter-efficient tuning and reducing computational resources.",
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+ "type": "text",
+ "text": "6.1 Homogeneous Transferable Tabular Model",
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+ "text": "Adapting a tabular model from another domain with different distributions is investigated in the field of unsupervised domain adaptation before the era of deep learning. One representative method is the biased regularization, which",
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+ "text": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX",
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+ {
+ "type": "image",
+ "img_path": "images/66966f1b48254c69ec49709de32610ec85ebc14caea1a2cea39c1e73a8debf60.jpg",
+ "image_caption": [
+ "Figure 5: Illustration of homogeneous transferable tabular methods. The pre-trained model could be constructed from supervised learning or self-supervised learning, which includes masked language model, contrastive pre-training, and hybrid methods."
+ ],
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+ "type": "text",
+ "text": "minimizes the difference between the weights of the PTM and the target model, i.e.,",
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+ "type": "equation",
+ "text": "\n$$\n\\min _ {\\boldsymbol {W}} \\ell (\\boldsymbol {W}) + \\| \\boldsymbol {W} - \\boldsymbol {W} ^ {\\prime} \\| _ {F} ^ {2} = \\min _ {\\Delta \\boldsymbol {W}} \\ell \\left(\\Delta \\boldsymbol {W} + \\boldsymbol {W} ^ {\\prime}\\right) + \\| \\Delta \\boldsymbol {W} \\| _ {F} ^ {2}. \\tag {5}\n$$\n",
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+ "text": "$\\ell(W)$ is the loss function on the current weights $W'$ , and the regularize constraint the distance between the target model $W$ and the PTM weights $W'$ . We can reformulate the learning objective as learning the weights residual $\\Delta W$ . Biased regularization can be extended to the case where $f$ and $g$ are deep neural networks such as MLP, but it fails when the target model has a different architecture with the PTM. In this case, instead of matching two models through their weights, matching their predictions also helps. For example, twice learning [253] and knowledge distillation [257].",
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+ "type": "text",
+ "text": "Benefiting from the strong capacity of deep neural networks, some recent studies focus on pre-training a tabular model from unsupervised instances, and then adapting the model via fine-tuning the PTM on the target (even few-shot) labeled examples. This strategy could be applied in standard supervised learning or semi-supervised learning.",
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+ "text": "Supervised Pre-training Objectives. A straightforward way to incorporate the target variable into the pre-training is by using the input corruption as an augmentation for the standard supervised learning objective. [71] identifies practices to pre-train tabular deep learning models that can be universally applied to different datasets and architectures. They show that using the object target labels during the pre-training stage benefits the downstream performance and advocates several target-aware pre-training objectives.",
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+ "text": "Self-Supervised Pre-training Objectives. The self-supervised pre-training objectives can be mainly categorized into three categories, including the masked language model, contrastive pre-training, and hybrid methods.",
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+ "type": "text",
+ "text": "Masked Language Model (MLM). MLM is the unsupervised pre-training objective, where a random subset of features is masked for each sample, and the masked values are predicted in a multi-target classification manner [63]. VIME [48] estimates mask vectors from corrupted tabular data and reconstructs feature vectors for self-supervised learning. They use the trained encoder to generate multiple augmented samples for each data point by masking each point using several different masks and then imputing the corrupted values for each masked data point. SubTab [46] finds that reconstructing the data from the subset of its features rather",
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+ {
+ "type": "text",
+ "text": "than its corrupted version in an autoencoder setting can better capture its underlying latent representation. SEFS [221] reconstructs the original input based on a randomly selected subset of input features, and simultaneously estimates the gate vector that defines which features are selected or not. MET [223] uses a concatenation of representations for all features instead of averaging and uses adversarial reconstruction loss in addition to the standard loss.",
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+ "text": "Contrastive Pre-training. Contrastive pre-training uses data augmentations to generate positive pairs or two different augmented views of a given example, and the loss function encourages a feature extractor to map positive pairs to similar features. The key factor in contrastive learning is to generate positive and negative versions of a given instance $x_{i}$ . [70] utilizes CutMix [258] in the input space and Mixup [259] in the embedding space to obtain positive pairs, where other instances $x_{j \\neq i}$ are treated as negative ones. SCARF [47] generates a view for a given input by selecting a random subset of its features and replacing them with random draws from their respective empirical marginal distributions. STab [224] relies on two (or multiple) weight-sharing neural networks with different regularizations applied to a single input. By exploiting the stop-gradient operation technique, STab can model invariance with respect to more complicated regularizations while it will not collapse to an undesired trivial solution. DoRA [226] incorporates domain knowledge, training by intra-sample pretext task and inter-sample contrastive learning to learn contextualized representations. DACL+ [220], to overcome the reliance on a particular domain, uses Mixup noise to create similar and dissimilar examples by mixing data samples differently either at the input or hidden-state levels.",
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+ "type": "text",
+ "text": "Hybrid Methods. [222] explores several pre-training strategies including both supervised and unsupervised ones. It considers MLM as the unsupervised pre-training objective, and sets multi-label classification as the supervised pre-training objective. By fine-tuning the PTM with several choices, including those with frozen feature extractor or not, the paper observes that supervised pre-training leads to more transferable features in the tabular domain. LFR [227] conducts pretraining by learning to simultaneously reconstruct multiple randomly generated projection functions. It considers diverse data types to show the wide-ranging applicability of learning from randomness, including tabular, vision, and language. ReConTab [225] utilizes both self-supervised learning and semi-supervised learning. It uses regularization techniques for raw feature selection and leverages contrastive learning with labels to distill the most pertinent information for downstream tasks. [71] focuses on the setup with fully labeled tabular datasets to understand if pretraining helps tabular deep learning in a fully supervised setting and compares pretraining methods to the strong supervised baselines. They show that using the object target labels during the pertaining stage is beneficial for the downstream performance and advocate several target-aware pretraining objectives. [256] provides a systematic review and summarizes the recent progress and challenges of self-supervised learning for non-sequential tabular data.",
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+ "text": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX",
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+ "image_caption": [
+ "Figure 6: Illustration of heterogeneous transferable tabular methods. During pre-training on one or multiple datasets, most of the parameters in the PTM are trained. For downstream tasks, only a small subset of parameters is fine-tuned while the rest remain fixed."
+ ],
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+ "text": "6.2 Heterogeneous Transferable Tabular Model",
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+ "text": "The main intuition lies in the mapping $f$ and $g$ work in a similar fashion, i.e., predicting the labels with similar mechanisms. Therefore, the main idea to transfer knowledge is to match the target model with the well-trained one, over the weight space or the prediction space.",
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+ "text": "Early methods mainly focus on the feature-level heterogeneity between $f$ and $g$ . One main assumption is that there exists a shared set of features between the pre-trained task $\\mathcal{D}'$ and the target task $\\mathcal{D}$ , then we may directly copy the weights corresponding to the shared features from the PTM. Some methods extend bias regularization to deal with heterogeneous feature spaces by padding the weights with zero. OPID [260] is a one-pass learning approach, which only needs to scan each instance once and to deal with evolving streams. In the pre-training stage, OPID compresses important information of vanished features into functions of survived features, and in the adaptation stage, it is expanded to include the augmented features. ReForm [261] learns the meta-representation for each feature and based on which calculates the relationship between features in the meta-representation space. ReForm then bridges the feature space gap through optimal transport, which could be further used to transform classifiers with different features and classes.",
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+ "text": "A major advantage of neural models is that they are easily fine-tuned in new domains and learn reusable features. For example, as the deep PTM has the ability to extract generalizable features for a tabular task, reusing the knowledge from the PTM can utilize the strategies designed for visual and language domains. In detail, we can fix most of the parameters in the PTM and tune the remaining parts which only have limited parameters, for example, the linear probing or parameter-efficient fine-tuning.",
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+ "text": "Reuse PTM Pre-trained from One Dataset. These methods primarily focus on the difference between the pre-trained and down-streaming datasets. TabRet [72] utilizes masked autoencoding to make the transformer work in downstream tasks. To transfer pre-trained large language models to tabular tasks, ORCA [73] trains an embedder to align the source and target distributions. TabToken [64] focuses on improving the quality of the feature tokens, which are an important component in tabular deep models. TabToken leverages a conditional contrastive loss to improve the",
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+ "type": "text",
+ "text": "quality of learned embeddings and demonstrates enhanced transferability of deep learning models for tabular data.",
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+ "text": "Pseudo-Feature method [222] utilizes pseudo-feature models individually for each new feature. In detail, given one additional feature in a downstream dataset, it first pretrains a model on the upstream data without that feature. Then Pseudo-Feature fine-tunes the pre-trained model on downstream data to predict values in the column absent from the upstream data. Next, the fine-tuned model is used back in the upstream datasets to predict and assign pseudo-values of this feature. After supplementing the upstream dataset with the \"unseen\" feature in the downstream task, PseudoFeature pre-trains and transfers the feature extractor to the downstream task again. This method is computationally expensive in our broader feature space adaptation scenario. Reuse PTM Pre-trained from Multiple Datasets. XTab [230] aims to enhance the transferability of the transformer. They address the challenge of inconsistent column types and quantities among tables by utilizing independent features and federated learning to pre-train the shared component.",
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+ "text": "Another thread of method learns shared components such as attribute-agnostic transformation across datasets, which provides a good model initialization for partial parameters given a downstream task. [228] infers latent representations of each attribute and each response from a few labeled instances using an inference network. The attribute and response representations are enabled make predictions based on the task-specific properties of attributes and responses even when attribute and response sizes are different across tasks. DEN [229] uses a three-block architecture: a covariate transformation block followed by a distribution embedding block and then a classification block. They provide theoretical insights to show that this architecture allows the embedding and classification blocks to be fixed after pre-training on a diverse set of tasks. Meta-Transformer [231] leverages a frozen encoder to perform multimodal perception without any paired multimodal training data. In Meta-Transformer, the raw input data from various modalities are mapped into a shared space in meta learning [262], allowing a subsequent encoder with frozen parameters to extract high-level semantic features.",
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+ "text": "6.3 Reusing a Pre-trained Language Model",
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+ "text": "In some cases, the semantic meaning of features is available, making it natural to leverage pre-trained language models for tabular data. Typically, two types of semantic information can be derived from a tabular dataset $\\mathcal{D}$ . First, attribute names for each of the $d$ features, $\\mathcal{A} = A_{1},\\ldots ,A_{d}$ , provide useful context. Additionally, meta-information such as a textual description, denoted as meta_description, can further enhance understanding. The learning process is then formulated as:",
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+ "type": "equation",
+ "text": "\n$$\n\\hat {y} _ {i} = f \\left(\\boldsymbol {x} _ {i}, \\mathcal {A} \\mid \\mathcal {D}, \\text {m e t a} _ {\\text {d e s c r i p t}}\\right) \\tag {6}\n$$\n",
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+ "type": "text",
+ "text": "where the semantic information bridges the gap between feature spaces and facilitates knowledge transfer from pretrained tasks to downstream applications.",
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+ "type": "text",
+ "text": "Although pre-trained language models have demonstrated success in various domains, their application to tabular data remains limited due to the prevalence of numerical values and the scarcity of textual descriptions.",
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+ "text": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX",
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+ "img_path": "images/886d5839208dbb126a86d29a0904962b263e4a1c1cdbafa3fd800eb2090af5e9.jpg",
+ "image_caption": [
+ "Figure 7: Illustration of transferable tabular methods with a language model. The language model can be applied at various stages, including feature tokenization, feature engineering, and textual serialization."
+ ],
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+ "text": "Moreover, concerns about data privacy and security may further restrict access to semantic information. Consequently, language models are typically applied to tabular datasets only when textual context is sufficiently available.",
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+ "text": "Language Models for Feature Tokenization. When the feature space changes, language-based methods assume that semantic relationships exist between feature descriptions and rely on large-scale language models to capture these connections. For example, the feature \"occupation\" in one task may share semantic similarity with the feature \"organization\" in another, allowing feature-label relationships to be reused across different datasets. By extracting feature embeddings (tokens), tables of varying sizes can be transformed into a standardized set of tokens in a shared space. A pre-trained transformer then encodes transferable knowledge, aiding the fine-tuning process for downstream tasks.",
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+ "text": "TransTab [77] trains a tokenizer based on the words present in tabular data and incorporates both column descriptions and table cells as raw input to a gated transformer model. The model is pre-trained via self-supervised learning or supervised contrastive loss and is validated on tasks such as transfer learning and feature incremental learning. PTab [232] adopts a similar approach, learning contextual representations from multiple tokenized tabular datasets before fine-tuning for downstream tasks. UniTabE [182] encodes and fuses information from column names, data types, and cell values into a set of tokens, applying an encoder-decoder architecture with Transformer and LSTM components. It is pre-trained using Multi-Cell-Masking and contrastive learning, where a sub-vector of an instance is treated as a positive sample while other instances or their subsets are considered negatives.",
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+ "text": "CM2 [79] introduces a cross-table pre-training framework that integrates attribute names and feature values. CM2 uses transformers to process feature tokens and employs a prompt-based Masked Table Modeling (pMTM) self-supervised objective, where column names act as prompts to assist in predicting masked features. TP-BERTa [78] follows a similar approach but incorporates numerical discretization strategies and magnitude tokenization for feature encoding, fine-tuning smaller pre-trained language models such as RoBERTa [263] for tabular data prediction. Its pre-training objective includes supervised loss and magnitude-aware triplet loss as a regularizer.",
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+ "type": "text",
+ "text": "CARTE [233] utilizes a graph representation of tabular",
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+ "text": "data to handle heterogeneous feature spaces, transforming textual information from column names and entries into embeddings. A graph-attentional network is then applied to contextualize entries with column names and neighboring entries. CARTE is pre-trained on the YAGO3 knowledge base [264] by constructing graphlets for tabular data and employing contrastive loss, where the original graphlet and one truncated variant are positives, while other graphlets in the batch serve as negatives. The pre-trained CARTE model is subsequently fine-tuned for downstream tasks.",
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+ "text": "Language Models for Feature Engineering. Discriminative features enhance the effectiveness of subsequent tabular learning models. Binder [234] identifies task input components that are not directly answerable by a model and leverages LLMs to generate auxiliary features, particularly for knowledge grounding tasks. Given that discriminative features are often manually designed, CAAFE [265] explores the use of LLMs to generate auxiliary features based on task and feature semantics. The quality of these features is then evaluated using a general tabular model, TabPFN [89]. FeatLLM [266] enhances feature generation by incorporating example-based prompting, enabling LLMs to create new features based on textual descriptions. TaPTaP [235] is expected to capture a generic tabular data distribution after ongoing pre-training on a large-scale corpus of real-world tabular data, generating high-quality synthetic tables to support various applications on tabular data.",
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+ "text": "Language Models for Textual Serialization. A direct approach to incorporating pre-trained language models involves converting tabular data into a textual format, allowing LLMs to infer relationships between features and labels based on embedded expert knowledge. This concept has been validated in semantic parsing tasks [267], [268]. LIFT [236] and TabLLM [80] serialize tabular data by integrating feature names into text and combining them with task descriptions. This enables LLMs to treat tabular prediction tasks as text generation problems. LIFT fine-tunes models on the entire training set, while TabLLM employs few-shot learning for fine-tuning. UniPredict [237] constructs prompts using metadata, sample serialization, and task instructions, fine-tuning LLMs with confidence-weighted augmented labels predicted by an external model. The approach is validated on multiple in-distribution datasets.",
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+ "text": "Despite their advantages, textual serialization methods face challenges when the number of features increases, as prompts may become too large to fit within the model's context window. The effectiveness of LLMs in tabular data tasks remains constrained by the availability of semantic information and the capabilities of external tabular models. Further exploration of LLM-based methods will be discussed in the general tabular models in Section 7.",
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+ "type": "text",
+ "text": "6.4 Reusing a Pre-trained Vision Model",
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+ "text": "Given the success of deep neural networks (DNNs) in visual tasks, it is intuitive to leverage the strong recognition capabilities of pre-trained vision models for tabular data. Additionally, data augmentation strategies commonly used in image processing can be introduced after transforming tabular data into a visual format. Similar ideas have been explored in time series forecasting [269] and irregular time series classification [270].",
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+ "type": "header",
+ "text": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX",
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+ "type": "page_number",
+ "text": "16",
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+ "type": "image",
+ "img_path": "images/1cb1d949fe6300f2c81dade78b72b5132ea16551e7587e0e4a4975de40932726.jpg",
+ "image_caption": [
+ "Figure 8: Illustration of transferable tabular methods with a vision model. Tabular data can be transformed into images through dimensionality reduction, table reorganization, and the use of image markers."
+ ],
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+ "text": "The primary challenge lies in representing tabular instances in an image-compatible format. In natural images, neighboring pixels often share semantic relationships, whereas tabular data lacks inherent spatial structure. Features in a tabular instance are permutation-invariant, meaning that exchanging their order does not alter the instance's meaning. Various methods have been proposed to transform tabular data into visual representations, enabling the application of pre-trained vision models fine-tuned for tabular tasks. This subsection highlights different transformation strategies that transfer tabular datasets into images.",
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+ "text": "Dimensionality Reduction Transformation. Visualization strategies for tabular data naturally convert tables into images by embedding high-dimensional features into a lower-dimensional space. DeepInsight [238] projects tabular data into a 2D space using t-SNE and constructs images through convex hull analysis, applying translation, rotation, quantization, and normalization. REFINED [239] employs Bayesian Metric Multidimensional Scaling to preserve pairwise distances within the low-dimensional representation, ensuring that structurally similar features remain proximate in the transformed image.",
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+ "text": "Table Reorganization Transformation. A tabular dataset $\\mathcal{D}$ can be treated as a matrix and represented as a single-channel image or kernel. To enable visual PTMs to recognize meaningful spatial relationships, different strategies have been developed for structuring tabular data into images. Tabular Convolution (TAC) [240] arranges data samples into zero-mean square matrices (kernels) of odd integer dimensions. These kernels are then convolved with a fixed \"base image,\" and the resulting images are subsequently fed to a CNN for classification. Image Generator for Tabular Data (IGTD) [74] and TabEye [75] share a similar idea, generating an image for each data sample where pixel intensities correspond directly to feature values. These methods prioritize placing similar features in close proximity but struggle with high-dimensional tabular tasks. LM-IGTD [241] extends IGTD by incorporating stochastic feature generation to enhance robustness and generalization.",
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+ "type": "text",
+ "text": "Image Marker Transformation. Another approach involves encoding feature values as visual markers within an image. Super-TML [242] assigns feature values to predetermined positions within an image, effectively handling categorical and numerical datasets. Tab2Visual [76] normalizes tabular data and represents each instance as a row of multiple bars, each corresponding to a specific value. Each feature",
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+ "type": "text",
+ "text": "is assigned a unique color to enhance visual differentiation, while bar widths are proportional to feature magnitudes.",
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+ "type": "text",
+ "text": "By transforming tabular data into images, these methods enable the application of powerful pre-trained vision models to tabular prediction tasks, leveraging established deep learning techniques from the vision domain to enhance tabular model performance.",
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+ "type": "text",
+ "text": "7 FROM TRANSFERABLE TO GENERAL MODEL",
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+ "text": "The general model (also referred to as the tabular foundation model) represents an advancement over the transferable model. It extends the generalization capabilities of a pretrained tabular model to a variety of heterogeneous downstream tabular tasks, regardless of their diverse feature and class spaces, without requiring additional fine-tuning. In other words, given a pre-trained model $g_{\\Theta}$ , it can be directly applied to a downstream tabular task $\\mathcal{D}$ to predict the label of a test instance $x^{*}$ as follows:",
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+ "type": "equation",
+ "text": "\n$$\n\\hat {y} ^ {*} = g _ {\\Theta} \\left(\\boldsymbol {x} ^ {*} \\mid \\mathcal {D}\\right). \\tag {7}\n$$\n",
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+ "text": "Thus, the general model shares similarities with the transferable tabular model, but with a greater emphasis on the \"zero-shot\" ability, aims to construct highly adaptive architectures capable of handling a wide array of heterogeneous datasets simultaneously. Importantly, it does not require an Adapt function, which further reduces the computational cost of hyper-parameter tuning. The goal of the general tabular model is to achieve better generalization on downstream tabular datasets $\\mathcal{D}$ when compared to alternative strategies, such as training a tabular model directly on $\\mathcal{D}$ or adapting a transferable model.",
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+ "text": "Remark 6. Distinguishing between an advanced transferable tabular model, pre-trained on a wide range of heterogeneous tabular tasks, and the general tabular model can be challenging. Some transferable tabular models, based on auxiliary feature semantics, are able to predict labels for downstream test instances directly [80]. However, their prediction ability is constrained and typically applicable only in specific areas after fine-tuning [78], [233]. The general tabular model, on the other hand, is designed to handle a wider range of heterogeneous tabular tasks, sharing similar pre-training challenges with transferable models but without utilizing additional semantics. Fine-tuning a pre-trained general model is also an option for further performance improvements [93], [96].",
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+ "type": "text",
+ "text": "Pre-training has revolutionized domains such as vision and language [271], [84], but its adoption in tabular data remains limited due to the inherent heterogeneity of tabular datasets. Tabular datasets can vary significantly in both dimensionality (i.e., the number of columns) and the semantic meaning of each dimension, even within the same application. For example, different healthcare datasets may capture varying levels of detail and aspects of patient information. Even within the same feature entry (e.g., the $d$ -th column), the meaning can vary (e.g., \"age\" vs. \"height\"). This contrasts with vision and text data (within the same language), where different data sources typically share the same \"vocabulary\" (e.g., pixels, patches, or sub-words) and similar relationships between vocabulary \"elements\" (e.g., neighboring pixels",
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+ "text": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX",
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+ "Figure 9: Illustration of general methods. These methods handle inherent heterogeneity by improving the model's adaptability or homogenizing the diverse tabular formats. Once pre-trained, they can be directly applied to downstream tasks without fine-tuning."
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+ "text": "often share colors). The lack of shared vocabulary and relationships in tabular data makes it challenging to jointly train a model across multiple datasets, let alone apply a pre-trained model directly to new downstream tasks.",
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+ "text": "There are two main strategies to address the inherent heterogeneity in tabular datasets: improving the model's adaptability or homogenizing the diverse tabular formats. We categorize general tabular models into three parts based on their strategies for achieving generalizability. The first focuses on raw-feature-based approaches, among which TabPFN variants represent a rapidly evolving branch and are thus discussed separately. The third category encompasses semantic-based methods that leverage attribute and task semantics to unify heterogeneous tasks.",
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+ "text": "7.1 Raw-Feature-based General Models",
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+ "text": "To adapt a general tabular model to heterogeneous tabular datasets during the pre-training and fine-tuning stages, two main strategies can be used from the data-centric and model-centric perspectives. From the data-centric perspective, the general model may standardize tabular datasets into a homogeneous form. For instance, TabPTM [86] transforms all datasets into a uniform format using meta-representation to enable pre-training. The pre-trained model can then be applied directly to a downstream dataset or fine-tuned without introducing additional parameters.",
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+ "text": "Alternatively, from the model-centric perspective, the general model may improve adaptability by tailoring it to specific tabular tasks. HyperFast [87] adopts the concept of a Hyper Network [272] in meta-learning [273], where a mapping from the tabular dataset to the weights of a classifier is learned. This mapping can then be used to predict labels for unseen instances from the task. To address datasets with varying dimensions, HyperFast projects datasets into a fixed size using random projections. To overcome the slow weight generation speed, MotherNet accelerates HyperFast by modifying its architecture with Transformer-like modules [88].",
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+ "text": "7.2 TabPFN Variants",
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+ "text": "The TabPFN family of models [89], [91] leverages the incontext learning capabilities of transformers, directly predicting labels by adapting test instances according to the context of training examples. In the first version of TabPFN, an instance $\\boldsymbol{x}_i$ is padded to a fixed dimension (e.g., 100), and the features are projected to a higher dimension (e.g., $d'$ ) for further processing. The label $y_i$ is processed similarly and",
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+ "text": "added to the instance embeddings. The embeddings of all $N + 1$ instances, including training and test instances, are formulated into a set of $N + 1$ tokens with $d'$ dimensions. These tokens are processed through several layers of a Transformer, and the output token corresponding to the test instance is further predicted using a 10-way classifier. TabPFN is pretrained over synthetically generated datasets with structured causal models (SCM) [274] and Bayesian Neural Networks (BNNs) [275], [276], enabling the strong in-context learning ability, with the best checkpoint selected based on some real-world datasets. Due to the high complexity of transformers, TabPFN is limited to small-scale tasks, with suggested sizes of $N < 1000$ , $d < 100$ , and $C < 10$ .",
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+ "text": "TabPFN v2 introduces a specialized feature tokenizer to better handle heterogeneity. Specifically, each cell in the table is projected to a $k$ -dimensional vector using a shared mapping, and random position encoding vectors are added to differentiate features [277]. This results in a tensor of size $(N + 1) \\times (d + 1) \\times k$ when there is a single test instance. The label of each instance is processed similarly, and the mapped $k$ -dimensional token is concatenated with the instance tokens. A dummy label (e.g., the average of all labels) is used for the test instance since its label is unknown. A two-way attention mechanism is used, with each feature attending to the other features in its row and then attending to the same feature across its column [278]. The output token corresponding to the label of the test instance is further mapped to a 10-class classifier or regressor. Several improvements have been made in TabPFN v2, including increased context size ( $N < 10000$ , $d < 500$ ), automatic feature engineering, and post-hoc ensemble methods. [279] analyzes TabPFN from a bias-variance perspective, shedding light on its generalization capabilities. Various applications have also been explored, including tabular data generation [280], anomaly detection [281], data augmentation [282], and time series forecasting [283].",
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+ "type": "text",
+ "text": "The improvements of TabPFN (especially TabPFN v1) stem from several aspects.",
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+ "text": "Pre-training Improvements. TabForestPFN [284] extends TabPFN by pre-training In-Context Learning (ICL)-transformers on a new forest dataset generator that creates unrealistic datasets with complex decision boundaries. TabDPT [179] pre-trains the architecture on real-world datasets using self-supervised learning and retrieval objectives, making it suitable for both classification and regression tasks. APT [285] is pre-trained utilizing adversarial synthetic data generated by adaptive agents, which systematically modify the underlying data-generating distribution and deliberately challenge the model with diverse synthetic datasets to enhance its robustness and generalization capabilities. TabICL [286] integrates tree-based SCMs using XGBoost [130] to model complex interactions and employs curriculum learning by progressively increasing synthetic dataset sizes. Scalable Improvements. The efficiency of TabPFN is highly sensitive to context size, prompting strategies to enhance scalability and performance [39]. These include compressing training data into a compact learned representation using sketching [287] or prompt tuning techniques [288], [289],",
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+ "type": "text",
+ "text": "1. Some variants of TabPFN are not considered general tabular models, especially the latter parts, as they require additional fine-tuning steps. We place them in this subsection due to their strong relationship with TabPFN.",
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+ {
+ "type": "header",
+ "text": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX",
+ "bbox": [
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+ "page_idx": 17
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+ "type": "page_number",
+ "text": "18",
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+ {
+ "type": "text",
+ "text": "employing adaptive data selection methods to identify the most pertinent training examples for each test instance [290], [90], [179], [291], and replacing traditional quadratic attention with computationally efficient linear attention mechanisms [292] and state-space models (SSMs) [293].",
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+ "text": "Adaptation Improvements. Some approaches improve TabPFN's performance on downstream tasks by adapting the context [90] or fine-tuning specific parts of the model [96], [284], [290], [289]. TabICL [286] employs a column-then-row attention mechanism to construct fixed-dimensional embeddings of rows, which are subsequently processed by a transformer like TabPFN v1 to facilitate efficient in-context learning. EquiTabPFN [294] introduces self-attention across target components, ensuring that the arbitrary ordering of target dimensions does not influence model predictions, enhancing the performance of TabPFN v1 to some extent.",
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+ "text": "7.3 Semantics-based General Models",
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+ "text": "By leveraging the semantic structure of tabular data, such as column names, heterogeneous tasks can be projected into a shared language space. This allows a single language model, pre-trained on diverse tabular datasets, to handle unseen tasks in a unified manner. TabuLa-8B [92] fine-tunes a Llama 3-8B LLM for tabular data prediction (classification and binned regression) using a novel packing and attention scheme for tabular prediction. GTL [93] transforms tabular datasets into an instruction-oriented language format, facilitating the continued pre-training of LLMs on instruction-oriented tabular data, which demonstrates strong performance in few-shot scenarios. GTL-S [295] unlocks the potential of GTL from a scaling perspective, revealing that scaling datasets and prediction tasks enhance generalization. [94] extends GTL by incorporating retrieval-augmented LLMs for tabular data, combined with retrieval-guided instruction-tuning for LLMs. MediTab [243] uses a data engine that leverages LLMs to consolidate tabular samples to overcome the barrier across tables with distinct schema. MediTab aligns out-domain data with the target task using a \"learn, annotate, and refinement\" pipeline, enabling the pre-trained model to infer for arbitrary tabular input in the domain without fine-tuning.",
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+ "type": "text",
+ "text": "8 TABULAR ENSEMBLE METHODS",
+ "text_level": 1,
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+ "text": "Ensemble learning is a natural way to improve the generalization ability of multiple base learners by leveraging their diversity. Classical methods such as Random Forest [127] and AdaBoost [126], [296] employ bagging and boosting, respectively, by ensembling multiple decision trees. These methods have proven effective for tabular data, as they reduce bias/variance and improve robustness [297].",
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+ "text": "In deep tabular learning, ensemble methods can be categorized into two primary approaches: joint-training ensembles, where multiple sub-networks are aggregated within a single training pipeline, and post-hoc ensembles, where the predictions from multiple pre-trained deep tabular models are fused. One major challenge in ensembling deep tabular methods is computational efficiency, as training multiple deep models or sub-models can be computationally expensive and time-consuming.",
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+ "text": "8.1 Joint-Training Ensembles",
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+ "text": "Joint-training ensemble methods integrate diverse model architectures within a single training process to improve predictive performance while maintaining efficiency. These architectures often combine different types of models, such as linear and non-linear models [28] or tree-based and deep neural network-based approaches [63]. Tree-mimic methods leverage this concept by mixing predictions from multiple tree nodes to enhance robustness [60], [59], [193].",
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+ "type": "text",
+ "text": "To improve efficiency while maintaining predictive power, various techniques have been explored. Some approaches employ parameter-efficient ensembles, such as TabM [176], which uses MLPs as base learners and incorporates BatchEnsemble [298] to generate multiple diverse base learners efficiently. This prevents a large increase in the number of learnable parameters while maintaining model diversity. Similarly, BETA leverages pre-trained TabPFN by generating multiple base learners through additional parameter tuning [96]. Specifically, BETA learns multiple feature projections, feeding the projected training sets into TabPFN and aggregating the results while applying BatchEnsemble to reduce the number of additional learnable parameters.",
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+ "text": "Some hybrid approaches, such as LLM-Boost and PFN-Boost, have been developed to integrate large language models and TabPFN with gradient-boosted decision trees [299]. In these approaches, LLMs and PFN serve as the initial base learners, and additional base learners are sequentially trained in a boosting manner. This approach leverages the strong prior knowledge from LLMs and TabPFN while maintaining the scalability of gradient-boosted decision trees.",
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+ "text": "8.2 Post-Hoc Ensembles",
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+ "text": "Post-hoc ensemble (PHE) methods involve combining multiple trained models to improve robustness and accuracy. Bagging-based ensembles are one of the most direct post-hoc strategies, where usually multiple models trained with different random seeds are aggregated [33], [69]. Although this approach improves model robustness, it incurs high computational overhead. Some recent studies have demonstrated that LLM-based methods exhibit diverse prediction behaviors compared to deep tabular models that do not utilize attribute names [94]. This difference in prediction styles enhances their complementarity, making them ideal candidates for ensemble methods.",
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+ {
+ "type": "text",
+ "text": "Instead of explicitly training multiple models, perturbation-based approaches create diverse predictions from the same pre-trained model. One such method applies feature permutation with TabPFN, leveraging the fact that TabPFN is not fully feature permutation-invariant [89]. A perturbation-based ensemble can be formed by randomly permuting the feature order in both the training and test sets and making predictions multiple times, generating multiple diverse predictors without additional training costs. TabPFN v2 introduces additional perturbations to enhance diversity among several key factors, including variations in feature encoding, feature quantization, categorical feature shuffling, SVD-based feature compression, outlier removal, and power transformations such as the Yeo-Johnson transformation [91]. These randomly selected transformations create diverse",
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+ {
+ "type": "header",
+ "text": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX",
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+ "type": "page_number",
+ "text": "19",
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+ {
+ "type": "text",
+ "text": "prediction patterns, enabling effective ensemble learning without requiring multiple separately trained models.",
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+ "type": "text",
+ "text": "Another post-hoc ensemble strategy employed in TabPFN v2 is the use of Portfolio-Based Ensemble, where a fixed set of TabPFN configurations is used [91]. A greedy ensemble selection technique is then applied to learn optimal weights for aggregating the predictions of different configurations [300]. By combining multiple perturbed models, this method improves generalization without excessive training costs. Some methods apply ensemble techniques to TabPFN v1 to handle large datasets. For instance, TabPFN-Bagging [96], [301] divides large datasets into multiple context groups, with the final results averaged to mitigate variance. BoostPFN [301] treats TabPFN v1 as weak learners, where each weak learner uses a subset of the training data as context. This approach allows BoostPFN to outperform standard Prior Fitted Networks (PFNs) on large datasets.",
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+ "type": "text",
+ "text": "9 EXTENSIONS",
+ "text_level": 1,
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+ {
+ "type": "text",
+ "text": "In this section, we briefly introduce some extensions on deep tabular methods across different complex tasks.",
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+ "type": "text",
+ "text": "Clustering. Traditional clustering approaches often leverage enhanced distance metrics, such as the Gower distance [302], which is specifically designed for mixed data types, and interpretable prototypes, such as K-medoids. Recent advances in tabular data clustering have sought to integrate interpretability constraints with deep representation learning. For example, IDC [97] introduces a deep learning framework for general tabular data that predicts interpretable cluster assignments at both the instance and cluster levels. To address overlapping clusters, TableDC [98] integrates the Mahalanobis distance, which accounts for variance and correlation within the data. This method provides a similarity measure suitable for tables, rows, or columns in high-dimensional latent spaces.",
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+ "text": "Anomaly Detection. Anomaly detection in tabular data is crucial for identifying subtle irregularities in structured datasets, such as fraudulent transactions or equipment failures. While classical techniques like Isolation Forest [303] and Local Outlier Factor [304] remain foundational, recent developments have incorporated various methods to capture contextual relationships in high-dimensional data. For instance, [305] introduces a method that learns mappings that maximize mutual information between each sample and the part that is masked out, capturing the structural nuances of samples from a single training class. ADBench [99] provides a comprehensive tabular anomaly detection benchmark with 30 algorithms and 57 benchmark datasets. Additionally, large language models (LLMs) have also been employed for anomaly detection in tabular data [306].",
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+ "type": "text",
+ "text": "Tabular Generation. Tabular data generation has become an essential tool for synthetic data creation, privacy preservation, and addressing data scarcity. Traditional methods, such as Bayesian networks or GANs, focus on mimicking marginal distributions, while recent advancements emphasize preserving complex feature dependencies and semantic consistency. For instance, tabular diffusion models [307] iteratively refine synthetic data to capture subtle correlations in high-dimensional datasets, outperforming GANs in terms of data",
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+ "type": "text",
+ "text": "fidelity. [308] introduces high-order structural causal information as a natural prior knowledge and offers a benchmark framework for evaluating tabular synthesis models. Despite these advances, challenges remain in balancing realism with privacy, such as avoiding identity leakage in sensitive datasets, and scaling to heterogeneous data types. Hybrid neuro-symbolic models [309] bridge this gap to provide trustworthy synthetic data for downstream tasks.",
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+ "type": "text",
+ "text": "Interpretability. Traditional gradient-boosted decision trees (GBDTs) inherently provide interpretability through feature importance scores and decision path visualization. Frameworks such as XGBoost [130] and LightGBM [131] quantify feature importance using metrics like split frequency and information gain. SHAP values [310] enable instance-level explanations by decomposing model predictions into feature contributions. The additive nature of GBDTs allows for partial dependence plots [311] to visualize feature effects while controlling for interactions. NeC4.5 [253], a novel decision tree algorithm that integrates the comprehensibility of decision trees with the generalization ability of neural network ensembles. By training a neural network ensemble to generate a new training set, NeC4.5 enhances decision tree performance while maintaining interpretability.",
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+ "type": "text",
+ "text": "Recent deep models specifically designed for tabular data have introduced novel interpretability mechanisms. For example, NAMs [255] combine some of the expressivity of DNNs with the inherent intelligibility of generalized additive models. They learn a linear combination of neural networks that each attend to a single input feature, which are trained jointly and can learn arbitrarily complex relationships between their input feature and the output. TabNet [105] uses sequential attention with learnable feature masks, where each decision step explicitly selects a subset of features via sparse masking. The aggregated feature usage across steps provides global interpretability comparable to GBDT's feature importance. Subsequent variants, such as TabTransformer [63], enhance interpretability by visualizing cross-feature attention patterns. FT-Transformer [33] combines feature tokenization with explainable attention, while NODE [60], NODE-GAM [61] and DOFEN [312] generalize ensembles of oblivious decision trees, benefiting from both end-to-end gradient-based optimization and multi-layer hierarchical representation learning.",
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+ "type": "text",
+ "text": "Open-Environment Tabular Machine Learning. Research on distribution shifts in tabular data starts with domain-to-domain shifts [110], which are commonly categorized based on the availability of target domain data. When target data is available, transfer learning techniques such as unsupervised domain adaptation [313] and test-time adaptation [314] are widely used. These methods adapt model parameters using test-time inputs but rely on access to target distributions, which may not always be feasible. In contrast, when target data is unavailable, a more practical but challenging scenario, methods aiming to enhance robustness and generalization, using approaches such as domain generalization [315], domain robustness [316], [317], label robustness [318] or ensemble strategies [95]. TableShift [110] provides a detailed analysis of this scenario.",
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+ {
+ "type": "text",
+ "text": "Beyond domain-to-domain shifts, temporal shifts are more general and complex. TabReD [109] emphasizes the inherent temporality of real-world tabular data, advocating",
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+ {
+ "type": "header",
+ "text": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX",
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+ "type": "page_number",
+ "text": "20",
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+ "type": "text",
+ "text": "for temporal splits for training and testing. [319] further propose a refined training protocol focusing on temporal evaluation, significantly improving generalization across models. To address temporal shifts, it's critical to incorporate temporal information [319]. Drift-Resilient TabPFN [174] models temporal shifts with a secondary SCM, which specifies changes in the primary model parameters. [319] introduce a plug-and-play temporal embedding that effectively captures trend and periodicity patterns, providing an adaptive mechanism to mitigate the impact of temporal shifts. Under temporal shift conditions, most methods experience performance degradation, while TabM [95] exhibits relative robustness [109]. However, [319] demonstrate that with the refined training protocol and temporal embedding, methods such as ModernNCA [35] can regain competitiveness.",
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+ {
+ "type": "text",
+ "text": "Multi-modal Learning with Tabular Data. Text, such as feature names, can be effectively utilized to enhance tabular data learning, as discussed in Section 6. Here, we focus on interactions with the image modality, e.g., in healthcare, where medical images require specialized equipment and expert knowledge, often in tabular form, for accurate diagnosis [320]. To tackle challenges like large medical datasets and high annotation costs, MMCL [106] uses a contrastive self-supervised learning framework that integrates images and tabular data. CHARMS [107] transfers expert knowledge from tabular data to images, improving image predictions even without tabular data during inference, thus reducing reliance on costly expert annotations. TIP [321] proposes a self-supervised learning strategy with a tabular encoder for incomplete, heterogeneous data and a multimodal interaction module for inter-modality representation learning.",
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+ "type": "text",
+ "text": "Tabular Understanding. Tabular understanding involves comprehending the information contained within a table and can be broken down into several tasks. For example, Table Detection (TD) [322], [323] refers to identifying the region of the image that contains the table while Table Structure Recognition (TSR) [324], [325] involves the identification of the rows and columns to identify individual table cells, which aims to recognize the cellular structures of tables from table images by extracting the coordinates of cell boxes and row/column spanning information. Table Question Answering (TQA) [326], [327], [112] refers to providing precise answers from tables to answer a user's question. Traditional methods, whether OCR-based [328], [329], [330] or OCR-free [331], [332], [333], [334], [335], have made significant strides in TSR and TD, which are relatively simpler tasks.",
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+ "text": "More complex tasks, such as TQA, have also been the focus of considerable effort. For example, Donut [332] proposes a novel task and a synthetic document image generator to pre-train the model, reducing reliance on large-scale real document images. Monkey and TextMonkey [336], [337] utilize shifted window attention and use similarity measures to filter out redundant tokens. mPLUG-DocOwl [338] adapts mPLUG-Owl for OCR-free document understanding, while TabPedia [335] constructs low- and high-resolution vision encoders with a concept synergy mechanism for visual table understanding. [339] focuses on exploring various table representations and directly prompting LLMs to improve performance. Please refer to [112], [113] for more details.",
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+ "text": "10 DISCUSSIONS",
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+ "text": "In this section, we discuss several possible future directions for tabular machine learning, particularly in light of the significant potential demonstrated by tabular general/foundation models.",
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+ "text": "The Ability to Handle Dynamic and Open Environments. Tabular models, particularly foundation models, will increasingly need to operate in dynamic, real-world environments where data evolves over time [340]. One of the key challenges is dealing with imbalanced datasets [155], where certain classes may be underrepresented, and the distribution of data may shift over time [110]. As a result, models need to adapt to these changes and continue providing accurate predictions. Additionally, the emergence of new classes in the data may require the model to evolve and update its predictions in real-time [341]. This calls for methods that ensure tabular foundation models can accommodate evolving data, handling both new classes and changing distributions effectively.",
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+ "text": "The Coverage and Scope of Tabular Foundation Models. Current tabular foundation models have demonstrated strong performance on various unseen classification and regression tasks. However, several important questions remain about their capabilities. For instance, in addition to in-context learning [246], are there other prediction strategies that could be employed to further enhance the versatility and performance of tabular foundation models? Beyond classification and regression, can these models be extended to handle related tasks such as clustering, imputation, outlier detection, or even table-based question answering (QA)? Expanding the task scope could increase the model's utility in a wide range of applications. Furthermore, it is worth investigating whether there is a scaling law [342] for tabular foundation models. Currently, tabular checkpoints are relatively small compared to foundation models in other modalities, such as language or vision. Understanding the implications of scaling these models—particularly the trade-offs between model size and performance—will be crucial for their future development.",
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+ "text": "Will Foundation Models Always Help? While foundation models have demonstrated impressive generalization abilities, there are inherent trade-offs. Similar to ensemble learning, a single foundation model may provide an \"average\" predictive ability across tasks, potentially losing specialized expertise for specific tasks. To address this, a promising approach could be the development of a \"tabular model zoo\" [343], [344]. In this paradigm, different pre-trained models, potentially including models from other domains, could be combined for a specific tabular task. Given a new task, suitable pre-trained models could be selected, adapted if necessary, and integrated for improved performance.",
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+ "text": "Model Efficiency. In many real-world applications, tabular datasets are large and high-dimensional, posing significant challenges for both training and inference [345], [44]. One area of concern is how to handle extreme cases, such as when the data is exceptionally large or sparse. Foundation models should be able to scale effectively in these scenarios without sacrificing performance. Another issue is inference speed. In large-scale problems, timely predictions are essential, especially when deployed in real-time environments [292]. Opti-",
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+ {
+ "type": "header",
+ "text": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX",
+ "bbox": [
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+ },
+ {
+ "type": "page_number",
+ "text": "21",
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+ },
+ {
+ "type": "text",
+ "text": "mizing the inference process is therefore critical to ensure that predictions can be made quickly on large, complex datasets. Lastly, the computational resources required for training and deploying foundation models can be substantial [346]. Optimizing resource usage through methods such as model pruning, quantization, and efficient training algorithms will be important to ensure that these models remain practical and accessible for a wide range of applications.",
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+ "text": "Bridging the Gap Between Tabular Data and Other Modalities. Tabular data often coexists with other data modalities, such as images and text. One of the exciting challenges in the field is how to effectively integrate tabular data with foundation models from other domains [347]. Combining the strengths of tabular models with those of vision or language models could result in more powerful and versatile models capable of handling multimodal data. Exploring how to seamlessly integrate these modalities—whether through joint embeddings, cross-modal attention mechanisms, or other techniques—could unlock significant advances in tasks that require both structured tabular data and unstructured data sources like images or text.",
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+ "type": "text",
+ "text": "11 CONCLUSION",
+ "text_level": 1,
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+ "text": "Tabular data remains a cornerstone of real-world machine learning applications, and the advancement of deep learning has opened new possibilities for effective representation learning in this domain. In this survey, we present a comprehensive overview of deep tabular representation learning, covering its background, challenges, evaluation benchmarks, and the discussion between tree-based models and DNNs. We systematically categorize existing methods into three categories—specialized, transferable, and general models—based on their generalization capabilities. In addition, we discuss ensemble techniques, extensions, and some promising future directions, such as open-environment and multimodal tabular learning. We hope this survey serves as a valuable reference for understanding the current state of the field and inspires further progress in developing more robust and generalizable tabular learning methods.",
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+ {
+ "type": "text",
+ "text": "REFERENCES",
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+ "bbox": [
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+ "page_idx": 28
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+ "list_items": [
+ "R. Child, S. Gray, A. Radford, J. Wu, and D. Amodei, \"Scaling laws for neural language models,\" CoRR, vol. abs/2001.08361, 2020. 21",
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+ "content": "Representation Learning for Tabular Data: A Comprehensive Survey"
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+ "content": "Jun-Peng Jiang, Si-Yang Liu, Hao-Run Cai, Qile Zhou, Han-Jia Ye"
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+ "content": "Index Terms—Tabular Data, Representation Learning, Deep Tabular Learning, Tabular Machine Learning, Tabular Foundation Model"
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+ "content": "1 INTRODUCTION"
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+ "content": "Tabular data, characterized by structured rows and columns, is one of the most prevalent data formats in real-world machine learning applications, spanning diverse domains such as finance [1], healthcare [2], education [3], recommendation systems [4], and scientific research. In particular, AI for scientific research (AI4science) has increasingly relied on tabular data, as numerous prominent datasets—such as those from genomics [5], chemistry [6], and climate science [7], [8]—naturally adopt tabular forms."
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+ "content": "Tabular data inherently organizes information in a structured, table-like format. In this survey, we focus primarily on supervised tabular machine learning tasks, specifically classification and regression. Beyond their structured organization, tabular datasets frequently include heterogeneous attributes [9], encompassing numerical, categorical, or mixed data types that may be either dense or sparse. Additionally, many tabular datasets present quality challenges, such as noisy measurements, missing values, outliers, inaccuracies [10], and privacy constraints [11], all of which complicate the modeling process. The most common supervised tabular tasks are classification and regression, where the goal is to learn mappings from training data to discrete or continuous targets, respectively. As illustrated in Figure 1, each row represents an instance (with its corresponding label), while each column corresponds to a specific attribute or feature [12]."
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+ "content": "Figure 1: A brief introduction to tabular data and associated learning tasks. Each row represents an instance and each column corresponds to a specific attribute or feature, which can be numerical or categorical. The most common tabular machine learning tasks are classification and regression as shown in the right side of the figure."
+ },
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+ "content": "Ideally, learned mappings should generalize effectively, accurately predicting outcomes for new instances drawn from the same underlying distribution."
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+ "content": "Machine learning methods for tabular data have evolved significantly over the years [13], [14], [15], [16]. Recently, the rise of deep learning has profoundly impacted domains like computer vision [17] and natural language processing [18], where Deep Neural Networks (DNNs) extract semantic representations directly from raw inputs [19], [20], [21]. These learned representations have not only improved generalization but have also facilitated knowledge transfer across related tasks [22]. The flexibility of DNNs in modeling complex feature interactions and learning rich hierarchical"
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+ "content": "- J.-P. Jiang, S.-Y Liu, H.-R Cai, Q. Zhou, and H.-J. Ye are with School of Artificial Intelligence, Nanjing University, and National Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, 210023, China. E-mail: {jiangjp.liusy,zhouql.yehj}@lamda.nju.edu.cn, caihr@smail.nju.edu.cn"
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+ "content": "Figure 2: We organize existing tabular classification/regression methods into three categories according to their generalization capabilities: specialized (left), transferable (middle), and general (right) models. Specialized models focus on tasks where training and evaluation occur within the same data distribution. Transferable models are pre-trained on one or more datasets and subsequently fine-tuned on downstream tasks. General models, also known as tabular foundation models, extend this concept further, allowing direct application to downstream tasks without additional fine-tuning."
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+ "content": "Indeed, DNNs were applied to tabular data decades ago, initially targeting dimensionality reduction and visualization tasks [23], [24], [25], [26], yet they typically struggled to match tree-based methods on standard classification and regression problems. Later advances in DNNs have led to significant improvements across various tabular-related applications, such as click-through rate prediction [27], [28], anomaly detection [29], recommendation systems [30], and time series forecasting [31], [32]. Modern deep learning approaches, benefiting from better-designed architectures, optimized training strategies, high-quality representations, have revitalized DNN performance on tabular data, often rivaling or surpassing traditional tree-based models [33], [34], [35]. Given the wide variety of approaches emerging in deep tabular modeling, a systematic overview that revisits critical factors and current methodologies in representation learning for tabular data has become increasingly necessary."
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+ "content": "This survey begins by introducing the background of tabular data learning, highlighting the challenges involved and critically examining the advantages and limitations of utilizing DNNs compared to classical—particularly tree-based—methods [36], [37], [38], [39]. Given the observed instability of method performance across different tabular datasets, we also discuss comprehensive strategies for dataset collection, evaluation, and analysis, aiming to establish robust criteria for aggregating performance metrics across multiple datasets [40], [41], [42], [43]."
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+ "content": "We broadly categorize deep tabular methods into three types: specialized methods, transferable methods, and general methods, distinguished by the scope of datasets on which they are trained and deployed, as well as their corresponding"
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+ "content": "generalization capabilities (illustrated in Figure 2). Specialized tabular methods align closely with classical supervised models, typically trained and evaluated on data drawn from the same distribution. In contrast, transferable methods leverage knowledge from models pre-trained on one or multiple source datasets, subsequently fine-tuning these models on target datasets; the primary challenge here lies in addressing the heterogeneity between pre-trained sources and target tasks. The recently proposed general tabular methods—motivated by the remarkable \"zero-shot\" generalization abilities demonstrated by large language models (LLMs)—exhibit exceptional versatility. These general models can directly apply their learned representations to downstream tabular datasets without additional fine-tuning, achieving robust generalization due to advanced pre-training strategies. Although the generalization ability tends to increase from specialized to general models, it does not imply that specialized or transferable methods are less valuable; specialized models remain superior on large-scale datasets, and fine-tuning general models can further improve their predictive performance. Additionally, the first two types of methods provide foundational insights and valuable components that contribute significantly to advancements in general tabular models."
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+ "content": "For specialized methods, numerous designs have been proposed from diverse perspectives, and previous papers have often categorized these methods based primarily on their architectural characteristics or behaviors. Existing taxonomies [44], for example, group specialized methods into feature-preprocessing-based [33], [45], data-augmentation-based [46], [47], [48], [49], MLP variants [50], [34], specialized DNN architectures [51], [52], [53], [54], [55], [56], [57], [58], tree-mimic approaches [59], [60], [61], token-based tech"
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+ "content": "niques [62], [63], [33], [64], [65], regularization-driven methods [66], [67], and neighborhood-based strategies [68], [69], [35]. However, such categorizations can appear scattered, making it difficult to connect the core ideas between methods placed in distinct groups. In contrast, this survey introduces a hierarchical taxonomy based on the key aspects of tabular data—features, samples, and objectives—providing a cohesive organizational framework. Our approach emphasizes detailed strategies for obtaining high-quality representations at both feature- and sample-levels. This unified perspective helps bridge core ideas across diverse methods, facilitating clearer comparative discussions and potentially guiding the design of future, more advanced tabular models."
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+ "content": "Instead of training a model from scratch on a single tabular dataset, transferable models leverage knowledge encoded in a pre-trained model from another dataset, which can significantly enhance the training process, especially when data or computational resources for the target task are limited. A major challenge in transferring knowledge across tabular tasks lies in the inherent heterogeneity between the source and target datasets, particularly differences in their feature and label spaces. In this survey, we adopt a broad perspective on transferable tabular models, categorizing methods based on the sources of their pre-trained knowledge. Specifically, we discuss models pre-trained on homogeneous tabular domains, such as self-supervised methods with additional pre-training steps on the target dataset itself [70], [71]; models pre-trained across heterogeneous tabular domains [72], [73], [64]; and methods transferring knowledge from other modalities, such as vision-based pre-trained models [74], [75], [76]. Additionally, since incorporating attribute semantics (when available) is a common strategy for bridging heterogeneous attribute spaces across tabular datasets [77], [78], [79], we also explore approaches leveraging language models in the final category. In particular, we further organize these language model-based strategies according to the methods they use to extract knowledge and the types of language models involved—ranging from small-scale language models to Large Language Models (LLMs) [80], [81], [82], [83]."
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+ "content": "Inspired by recent advancements in foundation models from vision and language domains [84], [85], general models—also known as tabular foundation models—expand the concept of transferable tabular models by enabling direct application to downstream tasks without additional fine-tuning. This capability, commonly referred to as the model's \"zero-shot\" ability, significantly enhances the model's usability across diverse tabular datasets. In contrast to transferable models, which primarily focus on bridging knowledge gaps between source and target datasets, general models aim to construct highly adaptive architectures capable of handling a wide array of heterogeneous datasets simultaneously. We categorize these general models based on the strategies used to achieve adaptiveness across diverse tabular tasks, specifically examining adaptations from both data-centric [86] and model-centric perspectives [87], [88]. Furthermore, we discuss critical branches of general tabular models in detail: the TabPFN variants leveraging in-context learning [89], [90], [91], and methods utilizing attribute and task semantics to unify heterogeneous tasks within a common representation framework [92], [93], [94]."
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+ "content": "Additionally, ensemble methods [95], [96], [91] are in"
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+ "content": "produced, which improve the generalization ability based on the strengths of multiple tabular models. Finally, we briefly overview other relevant extensions of tabular learning, including clustering [97], [98], anomaly detection [99], [100], [101], data generation and imputation [102], [103], [104], interpretability [63], [105], [61], multimodal learning [106], [107], open-environment tabular machine learning [108], [109], [110], [111], and tabular understanding [112], [113]. By summarizing the state of the field and identifying open challenges, we aim to guide future research and applications in tabular data representation learning."
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+ "content": "2 BACKGROUND"
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+ "content": "This section presents the (supervised) tabular machine learning task, including the notation of tabular data learning, the history of tabular data, the challenges of learning from tabular data, evaluation metrics, and tabular benchmarks."
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+ "content": "2.1 Learning with Tabular Data"
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+ "angle": 0,
+ "content": "A supervised tabular dataset is formatted as \\(N\\) examples and \\(d\\) features/attributes corresponding to \\(N\\) rows and \\(d\\) columns in the table. An instance \\(\\pmb{x}_i\\in \\mathbb{R}^d\\) is depicted by its \\(d\\) feature values. Assume \\(x_{i,j}\\) as the \\(j\\)-th feature of instance \\(\\pmb{x}_i\\), it could be a numerical (continuous) one \\(x_{i,j}^{\\mathrm{num}}\\in \\mathbb{R}\\), like the temperature of a region or the density of the object. \\(\\pmb{x}_i\\) can also be a categorical (discrete) value \\(x_{i,j}^{\\mathrm{cat}}\\), like one of multiple colors, the location of a person, or even some textual descriptions of the instance. Each instance is associated with a label \\(y_i\\), where \\(y_i\\in \\{1, - 1\\}\\) in a binary classification task, \\(y_i\\in [C] = \\{1,\\dots ,C\\}\\) in a multi-class classification task, and \\(y_i\\in \\mathbb{R}\\) in a regression task."
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+ "content": "Remark 1. Ordinal regression [114], [115], also called ordinal classification, is a type of regression analysis used to predict an ordinal variable. It can be considered an intermediate problem between regression and classification. However, this survey primarily focuses on standard classification and regression tasks and does not specifically discuss ordinal regression."
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+ "content": "Given a tabular dataset \\(\\mathcal{D} = \\{(x_i, y_i)\\}_{i=1}^N\\), we aim to learn a mapping \\(f\\) on \\(\\mathcal{D}\\) that maps \\(x_i\\) to its label \\(y_i\\). In other words, the model predicts \\(x_i\\) with \\(\\hat{y}_i = f(x_i)\\). The general objective learning \\(f\\) follows the structural risk minimization:"
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+ "content": "\\[\n\\min _ {f} \\sum_ {\\left(\\boldsymbol {x} _ {i}, y _ {i}\\right) \\in \\mathcal {D}} \\ell (y, \\hat {y} _ {i} = f \\left(\\boldsymbol {x} _ {i}\\right)) + \\Omega (f). \\tag {1}\n\\]"
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+ "content": "\\(\\ell (\\cdot ,\\cdot)\\) measures the discrepancy between the predicted label \\(\\hat{y}_i\\) and the true label \\(y_{i},e.g.\\), cross-entropy in classification and mean square error in regression. \\(\\Omega (\\cdot)\\) is the regularization on the model, which restricts the complexity of \\(f\\). We expect the learned \\(f\\) is able to extend its ability to unseen instances sampled from the same distribution as \\(\\mathcal{D}\\)."
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+ "angle": 0,
+ "content": "Tabular methods differ in their strategies to implement \\( f \\). The \"dummy\" approach makes predictions based on training labels \\( \\{y_i\\}_{i=1}^N \\) directly, which outputs the major class in the training set for classification and the average of all labels for regression, respectively."
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+ "angle": 0,
+ "content": "In a \\(C\\)-class classification task, classical parametric methods implement \\(f\\) with a linear mapping, i.e., \\(f(\\pmb{x}_i) = \\pmb{W}^\\top \\pmb{x}_i + \\pmb{b}\\), where the classifier \\(\\pmb{W} \\in \\mathbb{R}^{d \\times C}\\) and \\(\\pmb{b} \\in \\mathbb{R}^C\\)"
+ }
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+ "angle": 0,
+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "4"
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "is the bias. With different loss functions, we can implement Logistic Regression, SVM, or even AdaBoost. In contrast, non-parametric methods implement the prediction via \\( f(\\pmb{x}_i) = f(\\pmb{x}_i, \\mathcal{D}) \\), depending on the whole training set. For example, KNN searches neighbors in the training set \\( \\mathcal{D} \\) with the \\( K \\) smallest distance w.r.t. \\( \\pmb{x}_i \\). KNN can be viewed as a specific label smoother, with a dynamic local region for every instance. [116] links KNN and Random Forest from their ways of smoothing training labels in their predictions."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Deep tabular methods implement \\( f \\) with a deep neural network, e.g. Most deep learning models could be decomposed into two parts, i.e., \\( f(\\pmb{x}_i) = \\pmb{W}^\\top \\phi(\\pmb{x}_i) + \\pmb{b} \\). Similar to the linear model, \\( \\pmb{W} \\) and \\( \\pmb{b} \\) are the components of linear classifier, with \\( \\pmb{W} \\in \\mathbb{R}^{d' \\times C} \\). \\( \\phi \\) maps the input vector \\( \\pmb{x}_i \\) into the \\( d' \\) dimension space, which extracts semantic embeddings for the given tabular input. \\( \\phi \\) could be implemented with MLP or residual network."
+ },
+ {
+ "type": "title",
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+ "content": "2.2 History of Tabular Data"
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+ "content": "Historically, classical machine learning tasks were predominantly formulated with tabular data, or datasets readily transformed into a tabular representation without explicitly designating them as \"tabular.\" In early literature, the term \"tabular\" typically referred to tables within relational databases [117], CSV files on the web [118], or tables in documents [119]. Relevant tasks included table extraction [120], parsing [121], understanding [122], and discovering association rules [123]. With the expansion of machine learning applications into other modalities such as images, texts, audio, and video, the classical vector-based data representations have come to be explicitly termed \"tabular data.\""
+ },
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+ "content": "Early statistical approaches such as linear regression, logistic regression, linear discriminant analysis, and K-Nearest Neighbors (KNN) predate artificial intelligence. Classical learning methods further expanded across various paradigms, including decision trees [124], [125], multi-layer perceptrons (MLPs), support vector machines (SVMs), and nearest centroid classifiers [5], [14]. Ensemble methods enhanced predictive performance by aggregating outputs from multiple base learners [126], [127]. More recently, gradient boosting frameworks [128], [129], such as XGBoost [130], LightGBM [131], and CatBoost [132], have become prominent due to their effectiveness and efficiency in tabular data applications and competitions [133], [134]."
+ },
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+ "content": "With the development of deep learning, DNNs were applied to tabular classification and regression tasks decades ago, utilizing architectures such as stacked Restricted Boltzmann Machines and denoising autoencoders [135], [136], [137]. Early representation learning efforts primarily focused on dimensionality reduction and data visualization tasks [23], [24], [25], [26], yet these models struggled to surpass traditional tree-based methods in terms of generalization. However, advancements in neural network architectures and representation learning strategies have recently led to promising results in related tabular domains, including click-through rate prediction [27], [28], anomaly detection [138], [29], recommendation systems [139], [30], and time series forecasting [31], [140], [32], [141]. Innovations such as convolutional layers and learnable feature embeddings have improved the ability of deep models to capture high-order"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "attribute relationships [142], [143]. While early deep tabular methods lagged behind ensemble tree-based models, recent techniques have demonstrated competitive or superior performance [33], [34], [35], affirming deep representation learning as a promising direction for tabular data modeling."
+ },
+ {
+ "type": "text",
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+ "angle": 0,
+ "content": "While several survey papers have been published [9], [144], the field of tabular data has witnessed remarkable progress over the past two years. On one hand, the emergence of new specialized methods has introduced significant shifts in the landscape, motivating the need for our comprehensive taxonomy. On the other hand, the rise of transferable and general approaches has greatly enhanced the generality and applicability of tabular data modeling, which has been overlooked in previous works."
+ },
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+ "content": "2.3 Challenges of Learning from Tabular Data"
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+ "content": "Different from other types of data sources, e.g., images and texts, there exist several challenges dealing with tabular datasets due to their characteristics."
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+ "content": "Heterogeneity of Features. Unlike continuous image data or token-based textual data, tabular datasets often contain both numerical and categorical attributes, each requiring distinct handling methods [9], [145]. Numerical features frequently exhibit varying ranges and distributions, necessitating normalization or scaling. Categorical features differ in cardinality and semantic interpretation, requiring encoding methods like one-hot vectors or embeddings. Consequently, tabular models must carefully handle these mixed data types to preserve the usability of each feature."
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+ "content": "Lack of Spatial Relationships. Tabular data inherently lacks spatial or sequential relationships that are naturally found in other modalities [74], [50]. The order of columns has no semantic or spatial meaning, making tabular data permutation-invariant regarding features. Moreover, standard tabular machine learning assumes rows are independently and identically distributed (i.i.d.), further eliminating temporal or sequential correlations present in data such as video or time series. This absence of inherent spatial or sequential structure challenges deep learning architectures traditionally designed to exploit such dependencies."
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+ "angle": 0,
+ "content": "Low-quality and Missing Data. Compared to image or text data, where contextual or spatial redundancies help manage missing or corrupted values, tabular data is more vulnerable to incomplete or erroneous entries [146], [147]. Missing values in tabular datasets can introduce significant biases and degrade prediction quality. Additionally, noisy or incorrect values can considerably affect model reliability. Data preprocessing steps, including data cleaning and imputation, become crucial to maintaining accuracy and robustness in tabular machine learning."
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+ "angle": 0,
+ "content": "Importance of Feature Engineering. Effective tabular models heavily depend on the quality of their input features [45], [148]. Unlike image or textual data, where DNNs inherently learn feature representations from raw data, tabular methods often require domain-specific knowledge and meticulous manual feature engineering. Identifying and modeling complex, nonlinear interactions among tabular features frequently demands sophisticated transformations and expert insight, significantly impacting the predictive performance of models [149]."
+ }
+ ],
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+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
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+ "type": "page_number",
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+ "content": "Class Imbalance. Tabular datasets frequently exhibit imbalanced label distributions, especially in classification tasks, where certain categories are underrepresented [150], [151]. Class imbalance complicates model learning, leading to biased outcomes toward majority classes and poor performance on minority classes. Specialized methods such as oversampling, undersampling, or tailored loss functions (e.g., focal loss [152]) are required to address this imbalance effectively. Evaluation criteria like the AUC or F1-score further help assess model quality in imbalanced settings. Recent research highlights differences between deep and classical models in handling imbalance, emphasizing the need for careful consideration [153], [154], [155], [41]."
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+ "content": "Remark 2. Class imbalance has long been a known issue in the tabular domain, even before the rise of deep learning [156], and methods such as SMOTE [157], [158] can easily be extended to deep learning methods during preprocessing. However, Current deep tabular methods primarily assume that the training and testing data come from the same distribution, even in cases involving class imbalance. In addition, some class imbalance methods in visual domain can be readily extended to the tabular data learning [159], [160]. Therefore, we do not delve into class imbalance in this survey."
+ },
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+ "type": "text",
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+ "angle": 0,
+ "content": "Scalability to Large Datasets. Tabular datasets can become large-scale and high-dimensional, presenting computational and generalization challenges [161]. With increasing dimensionality, the risk of overfitting increases, especially when the number of features significantly surpasses the number of samples. Consequently, efficient training algorithms, memory management strategies, and sufficient computational resources become essential. Effectively scaling tabular models to handle large datasets while maintaining generalization ability remains a challenging but critical research area [162]. Model Selection and Hyperparameter Tuning. Tabular models are particularly sensitive to hyperparameter settings [163], [164]. Selecting an appropriate model architecture and tuning hyperparameters, such as learning rate, layer depth, or number of trees, can be computationally expensive and time-consuming. Despite the advancement of automated machine learning (AutoML) techniques [165], [166], [167], efficiently identifying optimal configurations for deep tabular methods under practical constraints remains challenging and critical for achieving high predictive performance."
+ },
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+ "content": "Domain-Specific Constraints. Certain application domains, such as healthcare or finance, impose additional regulatory or ethical requirements on model development [168]. For example, healthcare applications must comply with privacy standards like HIPAA [169] and provide explainability to clinicians. Financial models similarly must adhere to fairness regulations and industry standards. These constraints can restrict algorithm selection, necessitate interpretable predictions, and require additional validation, explainability, and auditability procedures [170], [171], [172]."
+ },
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+ "content": "2.4 Evaluation of a Tabular Method"
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+ "angle": 0,
+ "content": "We present the evaluation of tabular methods, ranging from traditional to modern, to provide a comprehensive evaluation across different aspects. For a given model on a dataset \\(\\mathcal{D}\\),"
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+ "angle": 0,
+ "content": "we employ standard metrics that quantify the discrepancy between the predicted label \\(\\hat{y}_i\\) and the true label \\(y_i\\)."
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+ "content": "Evaluation on A Single Task. For classification tasks, Accuracy (or Error Rate) is commonly employed as the primary metric. AUC and F1 scores are further used to address imbalanced label distributions, while Expected Calibration Error (ECE) [173], [174] calculates the weighted average error of the estimated probabilities. All criteria are the higher, the better, except the error rate and ECE. For regression tasks, common metrics include Mean Squared Error (MSE), Mean Absolute Error (MAE), and Root Mean Squared Error (RMSE), with MAE and RMSE sharing the scale of the original labels. Lower values denote superior performance. Additionally, the coefficient of determination \\((\\mathbb{R}^2)\\) is employed, with higher values indicating a better fit."
+ },
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+ "angle": 0,
+ "content": "In tabular machine learning, the diversity of datasets makes it difficult for any single model to consistently excel across all scenarios. Therefore, evaluating models requires not only assessing their performance on individual datasets but also employing aggregated metrics that capture their overall effectiveness across multiple datasets."
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+ "content": "Evaluation on A Set of Tasks. Early research predominantly relied on Average Rank (Friedman Rank) [12], [39], often used in conjunction with Critical Difference Comparisons, to evaluate model performance across multiple datasets. Models are ranked per dataset based on a chosen metric (e.g., accuracy, AUC, RMSE), and the average rank is computed across datasets. To ensure statistical robustness, hypothesis tests were employed to assess the significance of ranking differences, providing a more reliable comparative analysis. For multiple comparisons, tests such as the Wilcoxon-Holm, Fredman, and Nemiyi tests are employed [175]. To address the potential degradation of average rank by poor performance on some datasets, the Probability of Achieving the Maximum Accuracy (PAMA) [12] is defined as the fraction of datasets in which a model attains the highest accuracy. An alternative to PAMA accounts for near-optimal performance, \\( P95 \\) quantifies the likelihood of a model attaining at least \\( 95\\% \\) of the maximum accuracy, which is computed as the ratio of datasets where the classifier achieves at least \\( 95\\% \\) of the maximum accuracy to the total number of datasets."
+ },
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+ "type": "text",
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+ "content": "As research progressed, more diverse evaluation metrics were introduced. The Arithmetic Mean of a chosen metric provides a direct comparison across datasets, but variations in the scales of evaluation metrics across datasets can distort results. To mitigate this issue, performance metrics are often normalized before aggregation, with normalized Accuracy applied to classification tasks and normalized RMSE (nRMSE) used for regression [36], [34]. Depending on the evaluation framework, Mean Normalized Error can be used, but its dependence on normalization can hinder independent optimization. To further address these limitations, the Shifted Geometric Mean (SGM) error was introduced, which aggregates errors multiplicatively, reducing sensitivity to extreme values and ensuring more stable cross-datasets/splits comparisons [34]."
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+ "content": "Beyond absolute performance, relative comparisons are also important. The Relative Improvement metric quantifies a model's performance gain over a baseline (e.g., a simple MLP), offering insight into efficiency relative to simpler alternatives [176]. More recently, drawing inspiration from the ELO rating system[177], [178], ELO-based evaluation has"
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+ "angle": 0,
+ "content": "been introduced [179], modeling model-to-model comparisons as pairwise competitions across datasets. The ELO Score iteratively adjusts rankings based on relative performance, providing a more dynamic, fine-grained assessment."
+ },
+ {
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+ "content": "2.5 Tabular Benchmarks and Datasets"
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+ "content": "This section introduces existing benchmarks and datasets, along with associated considerations for constructing the benchmarks and evaluation protocols."
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+ "content": "2.5.1 Popular Tabular Benchmarks and Datasets"
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+ "content": "We first introduce several benchmarks based on raw features constructed from various aspects. Then, we present datasets with rich semantics, following some tabular toolboxes and evaluation protocols."
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+ "content": "Standard Benchmarks. Methods for tabular data have preferences depending on the dataset, and evaluating them on limited datasets can be easily influenced by randomness or other factors. Therefore, it's important to consider various aspects to ensure a more comprehensive and reliable benchmark evaluation."
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+ "content": "A comprehensive benchmark should cover a diverse set of datasets to test the model's generalization capabilities across different tasks and feature types. The benchmark should include datasets from different task types, including binary classification, multi-class classification, and regression tasks. [12] evaluates 179 classifiers across 17 families on 121 datasets, concluding that Random Forest variants were the most likely to perform best overall. [50] explores MLPs with parameterized techniques, such as ensembling and data augmentation, over 40 classification datasets. Similarly, [33] demonstrates the effectiveness of MLPs, ResNets, and transformer-based models on 11 datasets. [36] conducts experiments on 45 datasets, investigating the differences between tree-based and DNN-based methods."
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+ "content": "The benchmark should cover datasets with varying sizes, including datasets with a large number of samples and features as well as smaller datasets. The diversity of dataset sizes helps evaluate the scalability and efficiency of different models. [39] includes 176 classification datasets and evaluate 19 methods, comprising 8 classical and 11 deep methods. In this study, the pre-trained TabPFN model [89] emerges as the top performer on average, even when limited to randomly sampled training sets of 3000 examples. However, limited trials for hyperparameter tuning and strict time constraints in [39] may have led to suboptimal evaluations for some deep tabular methods [180]."
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+ "type": "text",
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+ "content": "To ensure robustness and generalization, datasets from multiple domains should be included. Common domains for tabular data include healthcare, biology, finance, education, and physics. Additionally, some datasets are derived from other domains, such as image or speech data, by feature extraction. [181] evaluates attention mechanisms and contrastive learning methods across 28 tabular datasets, comparing their performance with traditional deep learning and machine learning approaches. [44], with a particular focus on DNN-based models, uses a benchmark of over 300 tabular datasets spanning a wide range of task types, sizes, and domains. A more diverse collection allows us to assess whether a tabular method can generalize across applications."
+ },
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+ "type": "text",
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+ "content": "Semantic-Enriched Datasets. In addition, recent research has also focused on evaluating tabular data with rich semantics, such as incorporating meta information related to tasks or integrating attribute names. UniTabE [182] introduces a 7TB dataset containing 13 billion tabular examples for tabular pre-training, covering domains with investing, time series analysis, finance, economics, and with numerical, categorical, text data types. CM2 [79] proposes OpenTabs for crosstab pre-training, which contains an extensive collection of large-scale tables with column name semantics, including approximately 46M tabular samples. TP-BERTa [78] filters the OpenTabs for datasets with at least 10,000 samples and no more than 32 features, resulting in 101 binary classification datasets and 101 regression datasets with about 10 million samples. GTL [81] curates a collection of 384 public tabular datasets from Kaggle, which includes 176 classification and 208 regression tasks spanning a wide range of industrial domains. TabLib collects a set of 627M tables totaling 69TiB, along with 867B tokens of context [183]. TabLib was extracted from numerous file formats, including CSV, HTML, SQLite, PDF, Excel, and others, sourced from GitHub and Common Crawl. T4 (The Tremendous Tablib Trawl) [92] takes account of the inscrutable statistics and call sheets with personally identifiable information in TabLib and filters TabLib into a collection of 4M tables with 2.1B rows."
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+ "content": "Among these benchmarks and datasets, the semantic-rich ones are primarily used for pre-training LLMs on tabular data, while the others are mainly employed for evaluating standard methods. Besides, some toolboxes implement methods over tabular data, including those for classical methods, as well as those for deep tabular methods [184], [185], [186], [187], [188]. To establish a comprehensive tabular benchmark, several factors need to be considered, including the range of datasets and data quality."
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+ "content": "Remark 3. Recent studies have proposed alternative perspectives for tabular evaluations, such as focusing on dataset age [42], leveraging expert-level feature engineering [43], and considering dataset version [44]. Studies have also highlighted generalization in open word environments in tabular datasets [43], [109], where the distributions of training, validation, and test sets differ significantly. More discussions are in Section 9. Incorporating diverse, high-quality datasets helps build a reliable benchmark for meaningful model comparisons."
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+ "content": "2.5.2 Evaluation Protocols"
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+ "content": "Given the strong sensitivity of tabular methods to data and the additional randomness in deep methods, robust evaluation is essential. Furthermore, due to the high computational cost of some methods, it is equally important to ensure evaluation efficiency."
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+ "content": "Model Selection. Model selection on the validation set involves both hyperparameter tuning and early stopping, which are essential for reliable evaluation. Due to the large number of hyperparameters in deep methods, automated methods like Optuna [189] are commonly used to explore hyperparameters through multiple trials [33], [69]. During tuning, models are evaluated on the validation split, while models can also be trained with multiple random seeds, providing more reliable evaluations. In each trial and the"
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+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
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+ "content": "final training, early stopping [190] often employed to prevent overfitting, and the epoch with the best validation performance is selected as the final model."
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+ "content": "Performance Evaluation. To assess generalization and prevent overfitting, models are typically evaluated using separate train/val/test splits, with a typical split ratio of \\(64\\% / 16\\% / 20\\%\\). However, such fixed splits may yield inconsistent results. With the rise of deep learning, researchers have proposed more robust evaluation protocols to better reflect model capabilities [191]. Two main approaches are commonly used: (1) fixing the data split and running multiple trials with different random seeds [54], [59], [105], [69], [62], [87], [33], [58], [192], [65], [71]; and (2) using cross-validation, where new train/val/test splits are generated in each fold [63], [89], [193], [68], [34]. A hybrid strategy combining both random seeds and cross-validation is also adopted [194]."
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+ "content": "Recent studies show that holdout-based hyperparameter tuning can be unstable and prone to overfitting to the validation set [195], [180]. [180] found it ineffective on most TabZilla [39] datasets and instead used 5-fold cross-validation for more robust hyperparameter selection. As a result, they found the key meta-feature findings reported in [39] no longer held. This observation was also discussed in [44], which further identified meta-features that have a greater impact on model performance. For small datasets, alternative strategies have been proposed [196], [197], [198]. However, this approach significantly reduces the efficiency of hyperparameter search. [199] showed that simply reshuffling data splits can often improve generalization, making holdout selection competitive with cross-validation while remaining more computationally efficient."
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+ "content": "3 FROM CLASSICAL TO DEEP METHOD"
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+ "content": "We present possible advantages of deep learning for tabular data, as well as the potential challenges of deep learning when compared with tree-based methods."
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+ "content": "3.1 Advantages of deep representation learning"
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+ "content": "Deep tabular models offer several advantages beyond performance when compared with classical methods."
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+ "content": "Ability to Model Complex Feature Interactions. DNNs are particularly adept at capturing high-order, non-linear interactions between features, which may be challenging for traditional models like linear regression or decision trees [51], [54]. By learning a hierarchical representation of features, DNNs allow low-level feature interactions to be captured in the initial layers, while higher-order interactions are identified in deeper layers. This ability to automatically learn complex relationships makes DNNs highly effective in capturing intricate dependencies within tabular data."
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+ "content": "End-to-End Learning. Unlike traditional machine learning methods, which often involve separate steps for feature engineering, preprocessing, and model tuning, DNNs can process raw features and automatically extract useful representations without complex manual transformations. This end-to-end learning approach reduces human bias and simplifies the workflow, making the process more efficient. DNNs are trained through gradient optimization, enabling"
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+ "content": "a unified, streamlined solution for complex tasks [33], [107]. Additionally, deep models support multi-task learning, allowing related tasks to benefit from shared representations, enhancing both performance and efficiency [200], [70], [49]. Integration with Other Modalities. Deep tabular methods excel in multi-modal pipelines, where tabular data is integrated with other modalities, such as images, audio, or text. In AI4science applications, for instance, tabular data might be combined with image data [106], [107] (e.g., in medical imaging applications) or time-series data [201], [202] (e.g., in forecasting tasks). DNNs are well-suited to model interactions between heterogeneous data types, improving the overall performance. By jointly learning from multiple data sources, DNNs enhance their ability to make more accurate and comprehensive predictions across domains."
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+ "content": "Flexibility with Dynamic Environments. DNN-based methods benefit from the flexibility of gradient-based optimization, which allows efficient and iterative training. This flexibility makes DNNs adaptable to changing objectives without significant modifications, unlike tree-based models that often require specialized methods for different tasks [9]. Moreover, DNNs excel in dynamic environments, such as real-time predictions, financial analysis, and decision-making systems, where feature relationships may shift. This adaptability makes them suitable for online learning or incremental training, where new data is continuously integrated without retraining from scratch [203], [204]."
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+ "content": "Long-Term Knowledge Transfer and Learning. DNNs are capable of long-term learning and knowledge transfer, which allows them to retain valuable knowledge gained from training on diverse tasks [205]. Once trained on a broad set of tasks, DNNs can transfer this knowledge to related domains, reducing the need for complete retraining [206]. This is especially advantageous in fields like AI4science, where a model trained on one type of scientific data can be adapted to other related domains, saving both time and computational resources. This ability to transfer knowledge across tasks is a key advantage of deep learning, enabling more efficient use of data and model capabilities over time."
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+ "content": "3.2 Debates between Tree-Based Methods and DNNs"
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+ "content": "Although deep tabular methods have shown great potential in learning semantic representations and constructing nonlinear predictors, their initial performance often struggles to surpass that of classical tree-based ensemble methods, such as Gradient Boosted Decision Trees (GBDT). Many studies still treat GBDT approaches as strong baselines [36], [39], and in some cases, the advantages of deep tabular methods diminish as the number of evaluation datasets increases."
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+ "content": "Several reasons contribute to why tree-based methods retain their advantages over DNNs in many tabular tasks:"
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+ "content": "Better Handling of High-Frequency Data. Tree-based methods, particularly GBDT models, are highly efficient at handling high-frequency data or dense datasets with many small variations [38]. These models build decision trees by recursively splitting the data at the most informative feature points, capturing both local and global patterns efficiently. DNNs, on the other hand, may not capture fine-grained patterns as effectively without extensive regularization or tuning [207], [208]. To address this limitation, [38] introduced frequency reduction as an inductive bias through"
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+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
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+ "content": "the addition of scaling layers, while [45] demonstrated that periodic activation functions can significantly enhance neural networks' ability to learn high-frequency functions."
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+ "content": "Natural Handling of Mixed Data Types. Tabular data often includes a combination of numerical, categorical, and ordinal features [9], [44], [209]. Tree-based models are particularly strong when working with mixed data types, as they can handle categorical features directly without requiring one-hot encoding or embeddings. This ability to work with raw categorical data simplifies the preprocessing pipeline significantly. DNNs, however, generally require encoding techniques (e.g., one-hot encoding or learned embeddings) for categorical features, adding complexity and potentially leading to suboptimal performance [63]."
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+ "content": "Lower Computational Requirements for Training and Inference. For certain tasks, tree-based models tend to be more computationally efficient than DNNs [33]. GBDTs and other decision tree-based models can train relatively quickly and are less computationally intensive than deep neural networks [210], [39]. This is especially true when the dataset is not massive or when the model needs to be trained and deployed rapidly. DNNs, on the other hand, often require significant computational resources (e.g., GPUs, longer training times) to achieve comparable performance, making them less ideal in resource-constrained environments [211], [88]."
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+ "content": "Robustness to Noisy and Missing Data. Tree-based models are generally more robust to noisy data and missing values. When training a decision tree, missing values can be handled through optimal splitting that accommodates absent data, and trees can effectively deal with noisy or inconsistent data points [36]. DNNs, in contrast, are more sensitive to noise and often require careful preprocessing or specific techniques (e.g., data imputation or noise filtering) to avoid performance degradation with noisy or missing data [65], [89]."
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+ "content": "Interpretability and Transparency. Tree-based models offer a significant advantage in terms of interpretability [60], [61], [105]. The decision-making process of models like GBDT can be easily visualized in the form of decision paths, and feature importance can be directly extracted [130], [132], [131]. This transparency makes tree-based models appealing in domains where model explainability is crucial, such as in finance, healthcare, and regulated industries. Although interpretability techniques like LIME [212] and SHAP [213] exist for DNNs, tree-based models still tend to be more intuitive and easier to explain, especially in complex decision-making environments. Recent works [214], [60], [59], [193] have sought to bridge this gap by enhancing neural network interpretability through emulation of tree-based model behaviors."
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+ "content": "Handling Outliers and Skewed Data. Tree-based methods are often better at handling outliers and skewed distributions in the data. When a feature exhibits extreme values or skewed distributions, decision trees are inherently less sensitive to such anomalies because they create splits based on feature ranges that naturally isolate outliers. This characteristic can make them more robust than DNNs, which may require specialized loss functions or techniques (e.g., robust scaling or outlier removal) to handle such data points [43], [109]."
+ },
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+ "content": "4 TAXONOMY OF SPECIALIZED METHODS"
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+ "content": "Similar to the evolution of deep learning, which progresses from specialized learning to transfer learning and ultimately to foundation models [244], we categorize deep tabular methods into three groups, as shown in Figure 2: specialized methods, transferable methods, and general methods. This classification reflects both the evolutionary development of deep learning techniques and the increasing generalization capabilities of these models."
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+ "content": "Specialized methods, being the earliest developed and most widely used category, will be our starting point for discussion. Tabular data consists of features (columns), samples (rows), and objectives (labels), which together define the structure and the task objectives. We emphasize detailed strategies for obtaining high-quality representations at both feature- and sample-level for the target task. Specifically, given the input data, according to the general learning objective in Equation 1, we consider how to transform the tabular input \\( x_{i} \\) (feature aspect), how to construct relationships between samples (sample aspect), how to design the objective \\( \\ell(\\cdot) \\) and regularize \\( \\Omega(\\cdot) \\) (objective aspect). In particular,"
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+ "content": "- Feature Aspect. We focus on how to transform the raw tabular input (in various forms) into intermediate representations. We consider two types of features: numerical and categorical. By explicitly modeling the relationships between the two features (e.g., feature importance and interactions), we are able to enhance the model's understanding of the input space."
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+ "content": "- Sample Aspect. In addition to features, we explore how to retrieve and utilize neighboring samples to capture intersample dependencies, thereby improving predictions. In order to improve the model's ability to make predictions, we explore the relationships between a target sample and its \"extracted neighbors.\""
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+ "angle": 0,
+ "content": "- Objective Aspect. We examine how to modify the loss function and overall objective to introduce inductive biases. By directly guiding the learning process with the target variables, we incorporate prior knowledge or task-specific preferences into the model, thereby improving its generalizability and interpretability."
+ },
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+ "type": "text",
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+ "content": "In specialized methods, we focus solely on learning from pure data, excluding feature semantics considered in transferable methods (in Section 6), as they leverage the capabilities of language models. Since specialized methods encompass a wide range of approaches, and feature-aspect methods are the most extensive part of them, we will first introduce sample-aspect methods and objective-aspect methods in the following subsections. In Section 5, we will provide a detailed introduction to feature-aspect methods."
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+ "content": "4.1 Sample-aspect Specialized Methods"
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+ "content": "Sample interaction methods take a retrieval-based approach, focusing on relationships between individual samples rather than features. In a tabular dataset, each sample \\( x_{i} \\) represents a row with \\( d \\) features, and the goal is to leverage relationships between a target sample and its \"extracted neighbors\" to improve predictions."
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+ "angle": 0,
+ "content": "The general form for the sample interaction methods can be expressed as:"
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+ "type": "equation",
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+ "content": "\\[\n\\hat {y} _ {i} = f \\left(\\mathcal {R} \\left(\\boldsymbol {x} _ {i}, \\mathcal {D}; \\Phi\\right)\\right), \\tag {2}\n\\]"
+ }
+ ],
+ [
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+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
+ },
+ {
+ "type": "page_number",
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+ "type": "table_caption",
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+ "angle": 0,
+ "content": "Table 1: The taxonomy of representation learning for tabular data. The shade color in the last column denotes the subcategory, which is consistent with Figure 3."
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+ "content": "| Algorithm Category | Reference |
| Specialized Methods | § 5Feature-aspect Methods | Feature Encoding | [33], [45], [64] |
| Feature Selection | [59], [60], [105], [61], [193] |
| Feature Projection | [52], [33], [34], [58] |
| Feature Interaction | [54], [62], [63], [55], [65], [49], [215] |
| § 4.1Sample-aspect Methods | Sample Interaction | [70], [216], [217], [192], [67] |
| Neighbor Retrieval | [218], [68], [69], [35] |
| § 4.2Objective-aspect Methods | Training Objective | [67] |
| Training Regularization | [219], [50], [66] |
| § 6Transferable Methods | Homogeneous | [63], [48], [70], [220], [46], [221], [222], [223], [47], [224], [225], [226], [227] |
| Heterogeneous | [228], [229], [222], [72], [73], [64], [230], [231] |
| Language Model | [77], [232], [182], [79], [78], [233], [234], [82], [83], [235], [236], [80], [237] |
| Vision Model | [238], [239], [240], [74], [75], [241], [242], [76] |
| § 7General Mehtods | Raw-Feature-based | [86], [87], [88] |
| TabPFN Variants | [89], [91] |
| Semantics-based | [92], [93], [94], [243] |
"
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+ "angle": 0,
+ "content": "Figure 3: The roadmap of deep representation learning tabular methods. We organize representative methods chronologically to show the concentration at different stages. Different colors of these methods denote the sub-categories."
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+ "content": "where \\(\\mathcal{D}\\) is the set of all samples (training data) available for retrieval or learning. \\(\\mathcal{R}(\\cdot)\\) is the sample interaction module, which retrieves or aggregates information from relevant samples in \\(S\\) for the target sample \\(\\boldsymbol{x}_i\\). \\(\\Phi\\) represents the learnable parameters of \\(\\mathcal{R}\\). \\(f(\\cdot)\\) is the prediction head that maps the aggregated information to the final output \\(\\hat{y}_i\\)."
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+ "content": "Sample aspect approaches can be broadly categorized into two main strategies. The first approach introduces the modeling of sample relationships \\(\\mathcal{R}\\) during representation training, allowing the model to learn better representations by capturing inter-sample dependencies. The second ap"
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+ "content": "proach is retrieval-based models, which directly predict outcomes by learning how to retrieve and utilize neighbors' relationships \\(\\mathcal{R}\\) when testing."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.503,
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+ "angle": 0,
+ "content": "Sample Interaction. These methods assist in representation learning by allowing the model to capture relationships between samples, which in turn helps generate a more robust representation during training. During testing, the model becomes more sensitive to each sample without interaction."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.504,
+ 0.899,
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+ ],
+ "angle": 0,
+ "content": "SAINT [70] introduces inter-sample attention beyond inter-attribute attention, which improves row classification by relating each row to others in the table. NPT [216]"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.075,
+ 0.032,
+ 0.415,
+ 0.045
+ ],
+ "angle": 0,
+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.908,
+ 0.034,
+ 0.924,
+ 0.044
+ ],
+ "angle": 0,
+ "content": "10"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.073,
+ 0.054,
+ 0.493,
+ 0.228
+ ],
+ "angle": 0,
+ "content": "extends this via non-parametric Transformers, whereas Hopular [217] employs Hopfield networks, sharing conceptual alignment with SAINT [70]. Unlike nearest-neighbor classification, the distance metric is learned end-to-end. Prompt [192] posits that the feature importance in tabular data is sample-dependent. During feature extraction, it treats the information between samples as prompts. PTaRL [67] identifies two issues in the representation of tabular data samples: entanglement and localization. It addresses these by modeling global sample relationships through prototype generation and representation projection, helping the model produce clear and consistent decisions."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.073,
+ 0.228,
+ 0.492,
+ 0.302
+ ],
+ "angle": 0,
+ "content": "Neighbor Retrieval. These methods construct high-quality contexts to aid prediction by retrieving valuable neighbors and designing efficient ways to utilize them based on the relationships between samples. The training data is used to assist during testing."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.077,
+ 0.302,
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+ 0.565
+ ],
+ "angle": 0,
+ "content": "DNNR [68] argues that a key advantage of neighbor-based methods is the model's transparency, meaning that the model's decisions can be explained by inspecting its components. It enhances predictive performance by incorporating local gradient estimation and Taylor series approximation into the KNN framework. TabR [69] proposes that, compared to purely parametric (e.g., retrieval-free) models, retrieval-based models can achieve superior performance while also exhibiting several practically important properties, such as the ability for incremental learning and enhanced robustness. It encodes all candidate samples and then employs an attention-like mechanism to retrieve the samples that aid in making predictions, as explored in [218]. ModernNCA [35] revitalizes the classic tabular prediction method, Neighbourhood Component Analysis (NCA) [245], by designing and incorporating deep learning architectures and strategies. The resulting method efficiently leverages neighboring samples for prediction."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Remark 4. The neighborhood-based approach closely resembles"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "bles the current in-context learning [246] mechanism. In particular, the in-context learning used in general models like TabPFN [89], [91] can aslo be considered a form of the neighborhood method. This concept of neighborhood not only helps in standard tasks, but also enhances transferable and general methods. For example, LoCalPFN [90] highlights that employing local linear regression can lead to more expressive decision boundaries, while utilizing local context allows performance to scale with the size of the training dataset."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "content": "4.2 Objective-aspect Specialized Methods"
+ },
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The general objective learning \\( f \\) follows the structural risk minimization as in Equation 1, where \\( \\ell \\) is the loss function to set the training objective between the prediction and the ground truth label. \\( \\Omega(\\cdot) \\) is the regularization on the model, which directs the objective or restricts the complexity of \\( f \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "In traditional machine learning, models often rely on explicit regularization techniques on \\(\\Omega\\) to ensure good generalization. Methods such as decision trees, support vector machines, and linear models typically incorporate regularization terms directly into the loss function to control model complexity and prevent overfitting. For example, in linear regression, regularization methods like L1 (Lasso) [247], L2"
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+ {
+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Figure 4: Illustration of feature-aspect methods, including feature encoding, feature selection, feature projection and feature interaction."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "(Ridge) [248], or Elastic-Nets [249] penalize large coefficients, effectively controlling the complexity of the model and helping to maintain a balance between bias and variance."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Objective-aspect methods in deep learning are an extension of these traditional regularization techniques, where inductive bias is introduced by adjusting the loss function \\(\\ell\\) or adding regularizers \\(\\Omega\\). In the training process, the goal is to leverage regularization on the model to improve predictions."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Remark 5. Pre-train methods such as homogeneous transferable tabular methods in Section 6 also change the loss function \\(\\ell\\) or the regularization \\(\\Omega\\) to help pre-training. We will discuss these methods later."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Objective-aspect approaches can be broadly categorized into two main strategies. The first approach involves training objectives, which enhance the model with a specialized ability. The second approach introduces a regularizer, allowing the model to learn strong generalized representations."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Training Objective. For training objectives, PTaRL [67] constructs prototype-based projection space and learns the disentangled representation around global prototypes. PTaRL uses a diversification constraint for representation calibration and introduces a matrix orthogonalization constraint to ensure the independence of global prototypes."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.503,
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+ ],
+ "angle": 0,
+ "content": "Training Regularization. For training regularization, RLNs [219] overcome the challenge of an intractable number of hyperparameters during training by introducing an efficient tuning scheme, which minimizes a new \"Counterfactual Loss.\" In RLNs, the regularization coefficients are optimized together with learning the network weight parameters. RLNs produce extremely sparse networks, thus providing more interpretable models and revealing the importance that the network assigns to different inputs. [50] introduces \"cocktails,\" dataset-specific combinations of 13 regularization techniques, showing that even simple neural networks can outperform tree-based architectures when optimized with these methods. TANGOS [66] introduces a regularization-based improvement. It regularizes neuron attributions to encourage neurons to specialize and become orthogonal to one another."
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.075,
+ 0.032,
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+ 0.045
+ ],
+ "angle": 0,
+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.908,
+ 0.034,
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+ 0.043
+ ],
+ "angle": 0,
+ "content": "11"
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "5 FEATURE-ASPECT SPECIALIZED METHODS"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.072,
+ 0.072,
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+ ],
+ "angle": 0,
+ "content": "Tabular data is characterized by a diverse set of features, including both categorical and numerical variables. The complexity of tabular data arises from the variety of feature types, their interrelationships, and the high dimensionality often present. Traditional methods often rely on manual feature engineering, using techniques such as encoding categorical variables and selecting relevant features to improve model performance and reduce overfitting."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.073,
+ 0.19,
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+ ],
+ "angle": 0,
+ "content": "As deep learning has evolved, these traditional techniques have been integrated and expanded upon. Deep tabular models are capable of automatically learning complex feature representations, reducing the need for explicit feature engineering. Feature-aspect methods, such as feature encoding, selection, projection, and interaction, are essential for transforming raw tabular inputs into more informative intermediate forms. These methods help improve a model's ability to capture intricate relationships between features, thereby enhancing its generalization capabilities."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "5.1 Feature Encoding"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Various encoding strategies have been explored for both categorical and numerical features in tabular data. Additionally, with the advancement of the attention mechanism, feature tokenization, similar to word embeddings in natural language processing, transforms all features into embeddings."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Categorical Encoding. Categorical variables represent types of data which may be divided into groups. Examples of categorical variables are race, sex, age group, and educational level [250]. The categorical features are usually transformed in an index (integer). The two most popular techniques are an Ordinal Encoding and a One-Hot Encoding."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.072,
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+ ],
+ "angle": 0,
+ "content": "Ordinal Encoding assigns each unique category a distinct integer value. This approach is useful when the categorical variable has an inherent order, such as \"low,\" \"medium,\" and \"high.\" The main advantage of Ordinal Encoding is its simplicity and efficiency, as it transforms the categorical variable into a single numeric column. However, it assumes that there is an ordinal relationship between the categories, which may not always be the case. For instance, if the categorical variable represents \"color\" with categories such as \"red,\" \"blue,\" and \"green,\" applying Ordinal Encoding would introduce an artificial order that does not reflect any meaningful ranking."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.072,
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+ ],
+ "angle": 0,
+ "content": "On the other hand, One-Hot Encoding creates a new binary column for each unique category in the original categorical variable. For example, for a variable \"color\" with three categories (red, blue, and green), One-Hot Encoding would generate three binary columns: \"is_red,\" \"is_green,\" and \"is_green,\" encoding red as \\((1,0,0)\\), blue as \\((0,1,0)\\) and green as \\((0,0,1)\\). Each column indicates the presence or absence of that particular category. One-Hot Encoding is useful for nominal categorical variables, where no order exists between the categories. While One-Hot Encoding avoids the assumption of ordinal relationships, it can lead to a high-dimensional feature space if the categorical variable has many unique values, which may result in increased computational costs and potential issues with overfitting."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.074,
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+ ],
+ "angle": 0,
+ "content": "In some cases, more advanced encoding techniques are used to address the limitations of these basic approaches."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.503,
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+ ],
+ "angle": 0,
+ "content": "For example, Target Encoding assigns each category a value based on the mean of the target variable for that category. This method can be useful when there is a strong relationship between the categorical features and the target. In Leave-one-out embedding, every category is replaced with the mean of the target variable of that category, which excludes the current row to avoid overfitting."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Numerical Encoding. For encoding, MLP-PLR [45] introduces two numerical encoding methods: Piecewise Linear Encoding (PLE) and Periodic Activation Functions. These encoding methods can be integrated with other differentiable layers (e.g., Linear, ReLU) to enhance performance. PLE produces alternative initial representations for the original scalar values and is based on feature binning. Periodic Activation Functions take into account the fact the embedding framework where all features are computed independently of each other forbids mixing features during the embedding process and train the pre-activation coefficients instead of keeping them fixed. [38] utilizes tools from spectral analysis, showing that functions described by tabular datasets often have high irregularity, and can be smoothed by transformations such as scaling and ranking to improve performance. They propose \"frequency reduction\" as an inductive bias during training."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Feature Tokenization. Feature tokenizer performs a similar role to the feature extractor in traditional models. It transforms the input features to embeddings [62], [33]. Since the feature representations of features are very sparse and high-dimensional, a common way is to represent them into low-dimensional spaces (e.g., word embeddings)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The general form for feature tokenization can be expressed as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.613,
+ 0.508,
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+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {T} _ {i, j} = \\boldsymbol {b} _ {j} + \\mathcal {T} \\left(x _ {i, j}; \\Psi\\right) \\in \\mathbb {R} ^ {t}, \\tag {3}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.504,
+ 0.534,
+ 0.925,
+ 0.607
+ ],
+ "angle": 0,
+ "content": "where \\(\\mathcal{T}(\\cdot)\\) is the feature tokenizer module, which transforms the input feature vector \\(\\pmb{x}_i\\in \\mathbb{R}^d\\) to a token embedding \\(T_{i,j}\\in \\mathbb{R}^t\\) . \\(t\\) is the dimension of token embedding. \\(\\pmb{b}_{j}\\) is the \\(j\\) -th feature bias. \\(\\mathcal{T}\\) can be implemented with different forms. \\(\\Psi\\) represents the learnable parameters of \\(\\mathcal{T}\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.503,
+ 0.608,
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+ ],
+ "angle": 0,
+ "content": "In AutoInt [62], both the categorical and numerical features are embedded into low-dimensional spaces, which reduces the dimension of the input features and meanwhile allows different types of features to interact with each other. The embeddings of categorical features are computed by multiplying the embedding matrix with the multi-hot vector, while a corresponding embedding vector represents numerical features. TabTransformer [63] embed each categorical feature into a parametric embedding of dimension \\(t\\) using Column embedding. An embedding vector is assigned to each feature, and a set of embeddings is constructed for all categorical features. Unlike TabTransformer, SAINT [70] proposes projecting numerical features into a \\(t\\)-dimensional space before passing their embedding through the transformer encoder. FT-Transformer [33] adapts the Transformer architecture for tabular data, where all features are transformed to embeddings and applies a stack of Transformer layers to the embeddings. Specifically, the numerical tokenizer is implemented as the element-wise multiplication \\(\\boldsymbol{T}_i^{\\mathrm{num}} = \\boldsymbol{b}_i^{\\mathrm{num}} + x_i^{\\mathrm{num}} \\cdot \\boldsymbol{W}_i^{\\mathrm{num}}\\), and the categorical tokenizer is implemented as the lookup table \\(\\boldsymbol{T}_i^{\\mathrm{cat}} = \\boldsymbol{b}_i^{\\mathrm{cat}} + \\boldsymbol{e}_i^T \\boldsymbol{W}_i^{\\mathrm{cat}}\\), where \\(\\boldsymbol{e}_i^T\\) is a one-hot vector for the corresponding categorical feature. Other transformer-based"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.075,
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+ "angle": 0,
+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "12"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "methods, like [65], [72], [230], [215], use the same feature tokenizer as FT-Transformer."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "5.2 Feature Selection"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.073,
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+ ],
+ "angle": 0,
+ "content": "The high dimensionality of tabular data often causes overfitting, where the model focuses on irrelevant features and neglects the important ones. Feature selection reduces the number of features, retaining only the most valuable information. This helps prevent overfitting, improves generalization, and reduces computational complexity."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Traditional tree-based models facilitate automatic feature selection by evaluating the impact of each feature on the target during the construction process. Decision trees utilize metrics such as information gain or the Gini index for feature selection, while ensemble methods like random forests determine feature importance by assessing each feature's contribution [251], [252], [253]. Recently, modern deep learning methods for tabular data often mimic trees' structures for feature selection."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "GrowNet [59] and NODE [60] primarily mimic ensemble techniques. Inspired by GBDT, GrowNet designs a framework for building DNNs with multiple weak learners, where each learner's input consists of the original features plus the penultimate layer output from the previous learner. NODE uses a differentiable Oblivious Decision Tree as the base model, applying Bagging within each layer and Stacking across layers in a multi-layered structure. To make GAM [254] scalable and effective, NODE-GAM [61] modifies NODE to be a GAM, allowing GAM to learn quick, nonlinear jumps that better match patterns in real data."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "TabNet [105] and GRANDE [193] focus more on how tree models handle features. TabNet not only retains the representation learning capabilities of DNNs through self-supervised learning, but also incorporates the interpretability of tree models and the benefits of sparse feature selection, with a model structure designed for both feature selection and computation. GRANDE argues that the hard splits used by tree models are a key advantage over deep models, and thus proposes a method for learning hard, axis-aligned tree ensembles using gradient descent. GRANDE combines the beneficial inductive bias of axis-aligned splits with the flexibility provided by gradient descent optimization."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "5.3 Feature Projection"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Feature projection methods aim to project the raw data into a middle form, enhancing the representation ability for later architectures. Feature projection methods can be broadly categorized into two main approaches: MLP variants and special designed architectures. These approaches aim to enhance the model's ability to represent complex features for underlying feature structures."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.073,
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+ "angle": 0,
+ "content": "MLP Variants. For model architecture, RTDL [33] investigates both ResNet-like and Transformer-based architectures tailored for tabular data, proposing simple yet effective adaptations of these widely-used deep models. In particular, the MLP architecture is constructed by stacking multiple blocks consisting of Linear layers, ReLU activations, and Dropout, which transform the raw tabular features into a fixed-dimensional hidden representation. A final linear layer is then used as the classification head. The paper highlights"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "an important insight: with proper hyperparameter tuning, even simple architectures like MLP and ResNet can achieve competitive performance on tabular benchmarks."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Another contemporaneous work [50] enhances the MLP architecture by equipping it with a comprehensive suite of modern regularization techniques. Instead of introducing architectural innovations, this study focuses on systematically exploring combinations of 13 different regularization methods to identify an effective \"regularization cocktail\" for plain MLPs. The results demonstrate two key findings: (i) a well-regularized vanilla MLP can significantly outperform many recent, specialized neural architectures designed for tabular data; and (ii) such MLPs can even surpass strong traditional machine learning models like XGBoost across a range of benchmarks. For a more comprehensive strategy, RealMLP [34] explores multiple aspects including preprocessing, hyperparameters, architecture, regularization, and initialization."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Special Designed Architectures. For units, motivated by the observation that normalization techniques are prone to disturbances during training, SNN [52] proposes the Scaled Exponential Linear Unit (SELU) to improve deep models for tabular data. NAMs [255] uses exp-centered (ExU) hidden units to improve the learnability for fitting jumpy functions. BiSHop [58] uses a dual-component approach, sequentially processing data both column-wise and row-wise through two interconnected directional learning modules. They use layers of generalized sparse modern Hopfield layers, a sparse extension of the modern Hopfield model with learnable sparsity."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "5.4 Feature Interaction"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.504,
+ 0.526,
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+ ],
+ "angle": 0,
+ "content": "Feature interaction methods aim to model relationships among features to enhance the representation power of deep learning models on tabular data. In tabular datasets, each sample \\( \\boldsymbol{x}_i \\in \\mathbb{R}^d \\) is described by \\( d \\) features, and the goal is to transform these raw features into enriched representations that improve predictive performance."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.504,
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+ 0.643
+ ],
+ "angle": 0,
+ "content": "The general form for feature interaction methods can be expressed as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.645,
+ 0.643,
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+ 0.66
+ ],
+ "angle": 0,
+ "content": "\\[\n\\hat {y} _ {i} = f \\left(\\mathcal {H} \\left(\\boldsymbol {x} _ {i}; \\Theta\\right)\\right), \\tag {4}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "where \\(\\pmb{x}_i\\in \\mathbb{R}^d\\) is the input feature vector for a single instance, \\(\\mathcal{H}(\\cdot)\\) is the feature interaction module, which transforms the input \\(\\pmb{x}\\) by capturing feature dependencies or generating higher-order feature interactions. \\(\\Theta\\) represents the learnable parameters of \\(\\mathcal{H}\\). \\(f(\\cdot)\\) is the prediction head that maps the transformed representation to the final output \\(\\hat{y}\\)."
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+ "content": "Feature interaction methods can be broadly categorized into two main approaches: the design of automatic feature interaction modules and the mining of implicit feature relationships. These approaches aim to enhance the model's ability to learn complex feature interactions and underlying feature structures within tabular data."
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+ "content": "Automatic Feature Interaction Modules. These methods do not assume specific feature types within the tabular dataset. Instead, they focus on improving the feature interaction process, enabling the model to learn complex, high-order feature relationships autonomously."
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+ "angle": 0,
+ "content": "DCNv2 [54] improves the learning of the model's feature interaction by improving the \"Cross Network\" structure. It"
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+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "employs low-rank methods to approximate feature crosses in subspaces and then integrates these subspaces using a gating mechanism. AutoInt [62] maps the original sparse high-dimensional feature vectors into a low-dimensional space and models high-order feature interactions by stacking interaction layers with a multi-head attention mechanism. Unlike AutoInt, the TabTransformer[63] only maps categorical features into contextual embeddings and feeds them into a Transformer model, while numerical continuous features are directly concatenated with the interacted contextual embeddings. When tabular data contains only numerical features, TabTransformer behaves in an MLP-like manner. Conversely, when the data contains only categorical features, TabTransformer operates similarly to AutoInt."
+ },
+ {
+ "type": "text",
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+ "angle": 0,
+ "content": "Implicit Feature Relationships. Methods in this category typically assume that features in tabular data can be abstracted into implicit types and that it is necessary to design a suitable feature learning process to adapt to the characteristics of different types of features."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "DANets [55] propose the existence of underlying feature groups in tabular data, where features within each group are correlated. They learn to group input features and perform further feature abstraction. SwitchTab [49] introduces the idea of extracting sample-specific \"Salient Features\" and sample-shared \"Mutual Information\" in tabular features. It leverages self-supervised learning to assist in learning feature representations. ExcelFormer [65] argues that while DNN assigns weights to each feature, it does not actively exclude irrelevant features. To address this, it introduces Semi-Permeable Attention for feature interaction, which allows features with lower information content to access information from more informative features while preventing highly informative features from being influenced by less relevant ones. AMFormer [215] proposes the hypothesis that arithmetic feature interactions are crucial for deep tabular models. Based on the Transformer architecture, it introduces components designed to extract both additive and multiplicative interaction information."
+ },
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+ "content": "6 FROM SPECIALIZED TO TRANSFERABLE MODEL"
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+ "content": "Instead of training a tabular model from scratch, learning based on a Pre-Trained Model (PTM) may increase the learning efficacy and reduce the resource and data requirement. For example, in a house prices prediction task, training a regressor in a certain area may benefit from a well-trained predictor from its neighborhood."
+ },
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+ "content": "Learning by reusing the PTM usually contains two stages. The first is the pre-training of a tabular model, from one or more upstream tasks. Given the PTM and a downstream task, an adaptation strategy is needed to transform the PTM to the target task or facilitate the learning of the target model. Formally, a well-trained model \\( g_{\\Theta} \\) is often available and can be leveraged to facilitate the training of \\( f_{\\theta} \\) over \\( \\mathcal{D} \\). Here, \\( g_{\\Theta} \\) is pre-trained on a dataset \\( \\mathcal{D}' = \\{(x_j', y_j')\\}_{j=1}^{N'} \\) with instances \\( x_j' \\in \\mathbb{R}^{d'} \\) and labels \\( y_j' \\in [C'] \\). To reuse expert knowledge in \\( g_{\\Theta} \\), an adaptation strategy is applied: \\( f_{\\theta} = \\text{Adapt}(f_{\\theta_0} \\mid \\mathcal{D}, g_{\\Theta}) \\), where \\( \\theta_0 \\) is the initialization of the model. The notation could also be extended to cases with more than one PTM. The main challenge to reuse one or more PTMs is to bridge the gap between the PTM and the"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "target tabular model [256]. We categorize PTMs into three kinds based on the source of PTM \\( g_{\\Theta} \\)."
+ },
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+ "type": "text",
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+ "angle": 0,
+ "content": "Homogeneous Transferable Tabular Model. First, the PTM may come from the same form of task (with \\( d' = d \\) and \\( C' = C \\), but with different distributions \\( \\operatorname{Pr}(\\mathcal{D}') \\neq \\operatorname{Pr}(\\mathcal{D}) \\) or model families \\( g \\neq f \\)). For example, those pre-trained from other domains [71], or those unlabeled instances [48], [70]."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "Heterogeneous Transferable Tabular Model. In addition, we consider a PTM pre-trained from a slightly different task with \\(\\mathcal{D}\\). In addition to the previous difference, the PTM \\(g_{\\Theta}\\) may differ from \\(f_{\\theta}\\) in feature dimension \\((d' \\neq d)\\) or target class set \\((C' \\neq C)\\), so the adaptation method \\(\\mathbf{Adapt}(\\cdot)\\) must handle such heterogeneity [64], [230]."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Cross-Modal Transferable Tabular Model. Moreover, the pre-trained model could also be constructed from another modality, such as vision and language domains. The cross-modality PTM is hard to be applied to the tabular prediction task in most cases, so auxiliary information from the tabular task like the semantic meaning of attributes (i.e., the attribute names) are usually assumed to be available in this case, where PTM like large language models may provide the latent semantic meanings as external knowledge [77], [73]."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The main limitation of the transferable tabular model is the assumption that the data distribution of the well-trained model should be similar to the distribution of the target model. For example in the previous house price prediction task, if the PTM is pre-trained in an area distance from the target area and targets diverse problems, it is hard to utilize the PTM in the target task [222]. Since different tabular tasks may vary in their distribution, feature, or classes, the general assumption is their exist a common \"dimension\" between the PTM and the target task. Only the distribution changes under the shared dimension and classes, or there exists an overlap between the feature or class spaces [230]. For example, in real-world applications such as healthcare, there are numerous medical diagnostic tables. These tables usually have some features in common such as blood type and blood pressure. For rare diseases with limited data, knowledge transfer from other diagnostic tables with overlapping features becomes beneficial [228]. When the feature/label semantics are available, two different tasks may be linked through the semantic space, and textual PTMs can be used to map the tabular instance to this space or facilitate the prediction in this space [80]."
+ },
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+ "type": "text",
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+ "content": "Pros and Cons of transferable Models. Learning with a well-trained tabular model has several advantages based on the knowledge encoded in the PTM. First, the training efficiency of the target model is improved and the model may converge fast, as the PTM may provide better initialization weights or optimization paths. Then, the target model will reduce the requirement on the data size, i.e., learning with a few-shot dataset. Training based on a PTM also reduces the number of learnable parameters, leading to parameter-efficient tuning and reducing computational resources."
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+ "content": "6.1 Homogeneous Transferable Tabular Model"
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+ "content": "Adapting a tabular model from another domain with different distributions is investigated in the field of unsupervised domain adaptation before the era of deep learning. One representative method is the biased regularization, which"
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+ "content": "Figure 5: Illustration of homogeneous transferable tabular methods. The pre-trained model could be constructed from supervised learning or self-supervised learning, which includes masked language model, contrastive pre-training, and hybrid methods."
+ },
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+ "type": "text",
+ "bbox": [
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+ "content": "minimizes the difference between the weights of the PTM and the target model, i.e.,"
+ },
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+ "content": "\\[\n\\min _ {\\boldsymbol {W}} \\ell (\\boldsymbol {W}) + \\| \\boldsymbol {W} - \\boldsymbol {W} ^ {\\prime} \\| _ {F} ^ {2} = \\min _ {\\Delta \\boldsymbol {W}} \\ell \\left(\\Delta \\boldsymbol {W} + \\boldsymbol {W} ^ {\\prime}\\right) + \\| \\Delta \\boldsymbol {W} \\| _ {F} ^ {2}. \\tag {5}\n\\]"
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+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "\\(\\ell(W)\\) is the loss function on the current weights \\(W'\\), and the regularize constraint the distance between the target model \\(W\\) and the PTM weights \\(W'\\). We can reformulate the learning objective as learning the weights residual \\(\\Delta W\\). Biased regularization can be extended to the case where \\(f\\) and \\(g\\) are deep neural networks such as MLP, but it fails when the target model has a different architecture with the PTM. In this case, instead of matching two models through their weights, matching their predictions also helps. For example, twice learning [253] and knowledge distillation [257]."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "Benefiting from the strong capacity of deep neural networks, some recent studies focus on pre-training a tabular model from unsupervised instances, and then adapting the model via fine-tuning the PTM on the target (even few-shot) labeled examples. This strategy could be applied in standard supervised learning or semi-supervised learning."
+ },
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+ "type": "text",
+ "bbox": [
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+ "content": "Supervised Pre-training Objectives. A straightforward way to incorporate the target variable into the pre-training is by using the input corruption as an augmentation for the standard supervised learning objective. [71] identifies practices to pre-train tabular deep learning models that can be universally applied to different datasets and architectures. They show that using the object target labels during the pre-training stage benefits the downstream performance and advocates several target-aware pre-training objectives."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Self-Supervised Pre-training Objectives. The self-supervised pre-training objectives can be mainly categorized into three categories, including the masked language model, contrastive pre-training, and hybrid methods."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Masked Language Model (MLM). MLM is the unsupervised pre-training objective, where a random subset of features is masked for each sample, and the masked values are predicted in a multi-target classification manner [63]. VIME [48] estimates mask vectors from corrupted tabular data and reconstructs feature vectors for self-supervised learning. They use the trained encoder to generate multiple augmented samples for each data point by masking each point using several different masks and then imputing the corrupted values for each masked data point. SubTab [46] finds that reconstructing the data from the subset of its features rather"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "than its corrupted version in an autoencoder setting can better capture its underlying latent representation. SEFS [221] reconstructs the original input based on a randomly selected subset of input features, and simultaneously estimates the gate vector that defines which features are selected or not. MET [223] uses a concatenation of representations for all features instead of averaging and uses adversarial reconstruction loss in addition to the standard loss."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "Contrastive Pre-training. Contrastive pre-training uses data augmentations to generate positive pairs or two different augmented views of a given example, and the loss function encourages a feature extractor to map positive pairs to similar features. The key factor in contrastive learning is to generate positive and negative versions of a given instance \\( x_{i} \\). [70] utilizes CutMix [258] in the input space and Mixup [259] in the embedding space to obtain positive pairs, where other instances \\( x_{j \\neq i} \\) are treated as negative ones. SCARF [47] generates a view for a given input by selecting a random subset of its features and replacing them with random draws from their respective empirical marginal distributions. STab [224] relies on two (or multiple) weight-sharing neural networks with different regularizations applied to a single input. By exploiting the stop-gradient operation technique, STab can model invariance with respect to more complicated regularizations while it will not collapse to an undesired trivial solution. DoRA [226] incorporates domain knowledge, training by intra-sample pretext task and inter-sample contrastive learning to learn contextualized representations. DACL+ [220], to overcome the reliance on a particular domain, uses Mixup noise to create similar and dissimilar examples by mixing data samples differently either at the input or hidden-state levels."
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+ "type": "text",
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+ "content": "Hybrid Methods. [222] explores several pre-training strategies including both supervised and unsupervised ones. It considers MLM as the unsupervised pre-training objective, and sets multi-label classification as the supervised pre-training objective. By fine-tuning the PTM with several choices, including those with frozen feature extractor or not, the paper observes that supervised pre-training leads to more transferable features in the tabular domain. LFR [227] conducts pretraining by learning to simultaneously reconstruct multiple randomly generated projection functions. It considers diverse data types to show the wide-ranging applicability of learning from randomness, including tabular, vision, and language. ReConTab [225] utilizes both self-supervised learning and semi-supervised learning. It uses regularization techniques for raw feature selection and leverages contrastive learning with labels to distill the most pertinent information for downstream tasks. [71] focuses on the setup with fully labeled tabular datasets to understand if pretraining helps tabular deep learning in a fully supervised setting and compares pretraining methods to the strong supervised baselines. They show that using the object target labels during the pertaining stage is beneficial for the downstream performance and advocate several target-aware pretraining objectives. [256] provides a systematic review and summarizes the recent progress and challenges of self-supervised learning for non-sequential tabular data."
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+ "content": "Figure 6: Illustration of heterogeneous transferable tabular methods. During pre-training on one or multiple datasets, most of the parameters in the PTM are trained. For downstream tasks, only a small subset of parameters is fine-tuned while the rest remain fixed."
+ },
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+ "content": "6.2 Heterogeneous Transferable Tabular Model"
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+ "content": "The main intuition lies in the mapping \\( f \\) and \\( g \\) work in a similar fashion, i.e., predicting the labels with similar mechanisms. Therefore, the main idea to transfer knowledge is to match the target model with the well-trained one, over the weight space or the prediction space."
+ },
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+ "type": "text",
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+ "angle": 0,
+ "content": "Early methods mainly focus on the feature-level heterogeneity between \\( f \\) and \\( g \\). One main assumption is that there exists a shared set of features between the pre-trained task \\( \\mathcal{D}' \\) and the target task \\( \\mathcal{D} \\), then we may directly copy the weights corresponding to the shared features from the PTM. Some methods extend bias regularization to deal with heterogeneous feature spaces by padding the weights with zero. OPID [260] is a one-pass learning approach, which only needs to scan each instance once and to deal with evolving streams. In the pre-training stage, OPID compresses important information of vanished features into functions of survived features, and in the adaptation stage, it is expanded to include the augmented features. ReForm [261] learns the meta-representation for each feature and based on which calculates the relationship between features in the meta-representation space. ReForm then bridges the feature space gap through optimal transport, which could be further used to transform classifiers with different features and classes."
+ },
+ {
+ "type": "text",
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+ "content": "A major advantage of neural models is that they are easily fine-tuned in new domains and learn reusable features. For example, as the deep PTM has the ability to extract generalizable features for a tabular task, reusing the knowledge from the PTM can utilize the strategies designed for visual and language domains. In detail, we can fix most of the parameters in the PTM and tune the remaining parts which only have limited parameters, for example, the linear probing or parameter-efficient fine-tuning."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Reuse PTM Pre-trained from One Dataset. These methods primarily focus on the difference between the pre-trained and down-streaming datasets. TabRet [72] utilizes masked autoencoding to make the transformer work in downstream tasks. To transfer pre-trained large language models to tabular tasks, ORCA [73] trains an embedder to align the source and target distributions. TabToken [64] focuses on improving the quality of the feature tokens, which are an important component in tabular deep models. TabToken leverages a conditional contrastive loss to improve the"
+ },
+ {
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+ "bbox": [
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+ "angle": 0,
+ "content": "quality of learned embeddings and demonstrates enhanced transferability of deep learning models for tabular data."
+ },
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+ "bbox": [
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+ "angle": 0,
+ "content": "Pseudo-Feature method [222] utilizes pseudo-feature models individually for each new feature. In detail, given one additional feature in a downstream dataset, it first pretrains a model on the upstream data without that feature. Then Pseudo-Feature fine-tunes the pre-trained model on downstream data to predict values in the column absent from the upstream data. Next, the fine-tuned model is used back in the upstream datasets to predict and assign pseudo-values of this feature. After supplementing the upstream dataset with the \"unseen\" feature in the downstream task, PseudoFeature pre-trains and transfers the feature extractor to the downstream task again. This method is computationally expensive in our broader feature space adaptation scenario. Reuse PTM Pre-trained from Multiple Datasets. XTab [230] aims to enhance the transferability of the transformer. They address the challenge of inconsistent column types and quantities among tables by utilizing independent features and federated learning to pre-train the shared component."
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+ {
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+ "angle": 0,
+ "content": "Another thread of method learns shared components such as attribute-agnostic transformation across datasets, which provides a good model initialization for partial parameters given a downstream task. [228] infers latent representations of each attribute and each response from a few labeled instances using an inference network. The attribute and response representations are enabled make predictions based on the task-specific properties of attributes and responses even when attribute and response sizes are different across tasks. DEN [229] uses a three-block architecture: a covariate transformation block followed by a distribution embedding block and then a classification block. They provide theoretical insights to show that this architecture allows the embedding and classification blocks to be fixed after pre-training on a diverse set of tasks. Meta-Transformer [231] leverages a frozen encoder to perform multimodal perception without any paired multimodal training data. In Meta-Transformer, the raw input data from various modalities are mapped into a shared space in meta learning [262], allowing a subsequent encoder with frozen parameters to extract high-level semantic features."
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+ "content": "6.3 Reusing a Pre-trained Language Model"
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+ "content": "In some cases, the semantic meaning of features is available, making it natural to leverage pre-trained language models for tabular data. Typically, two types of semantic information can be derived from a tabular dataset \\(\\mathcal{D}\\). First, attribute names for each of the \\(d\\) features, \\(\\mathcal{A} = A_{1},\\ldots ,A_{d}\\), provide useful context. Additionally, meta-information such as a textual description, denoted as meta_description, can further enhance understanding. The learning process is then formulated as:"
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+ "content": "\\[\n\\hat {y} _ {i} = f \\left(\\boldsymbol {x} _ {i}, \\mathcal {A} \\mid \\mathcal {D}, \\text {m e t a} _ {\\text {d e s c r i p t}}\\right) \\tag {6}\n\\]"
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+ "angle": 0,
+ "content": "where the semantic information bridges the gap between feature spaces and facilitates knowledge transfer from pretrained tasks to downstream applications."
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+ "content": "Although pre-trained language models have demonstrated success in various domains, their application to tabular data remains limited due to the prevalence of numerical values and the scarcity of textual descriptions."
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+ "content": "16"
+ },
+ {
+ "type": "image",
+ "bbox": [
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+ "content": "Figure 7: Illustration of transferable tabular methods with a language model. The language model can be applied at various stages, including feature tokenization, feature engineering, and textual serialization."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "Moreover, concerns about data privacy and security may further restrict access to semantic information. Consequently, language models are typically applied to tabular datasets only when textual context is sufficiently available."
+ },
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+ "type": "text",
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+ "content": "Language Models for Feature Tokenization. When the feature space changes, language-based methods assume that semantic relationships exist between feature descriptions and rely on large-scale language models to capture these connections. For example, the feature \"occupation\" in one task may share semantic similarity with the feature \"organization\" in another, allowing feature-label relationships to be reused across different datasets. By extracting feature embeddings (tokens), tables of varying sizes can be transformed into a standardized set of tokens in a shared space. A pre-trained transformer then encodes transferable knowledge, aiding the fine-tuning process for downstream tasks."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "TransTab [77] trains a tokenizer based on the words present in tabular data and incorporates both column descriptions and table cells as raw input to a gated transformer model. The model is pre-trained via self-supervised learning or supervised contrastive loss and is validated on tasks such as transfer learning and feature incremental learning. PTab [232] adopts a similar approach, learning contextual representations from multiple tokenized tabular datasets before fine-tuning for downstream tasks. UniTabE [182] encodes and fuses information from column names, data types, and cell values into a set of tokens, applying an encoder-decoder architecture with Transformer and LSTM components. It is pre-trained using Multi-Cell-Masking and contrastive learning, where a sub-vector of an instance is treated as a positive sample while other instances or their subsets are considered negatives."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "CM2 [79] introduces a cross-table pre-training framework that integrates attribute names and feature values. CM2 uses transformers to process feature tokens and employs a prompt-based Masked Table Modeling (pMTM) self-supervised objective, where column names act as prompts to assist in predicting masked features. TP-BERTa [78] follows a similar approach but incorporates numerical discretization strategies and magnitude tokenization for feature encoding, fine-tuning smaller pre-trained language models such as RoBERTa [263] for tabular data prediction. Its pre-training objective includes supervised loss and magnitude-aware triplet loss as a regularizer."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "CARTE [233] utilizes a graph representation of tabular"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.503,
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+ ],
+ "angle": 0,
+ "content": "data to handle heterogeneous feature spaces, transforming textual information from column names and entries into embeddings. A graph-attentional network is then applied to contextualize entries with column names and neighboring entries. CARTE is pre-trained on the YAGO3 knowledge base [264] by constructing graphlets for tabular data and employing contrastive loss, where the original graphlet and one truncated variant are positives, while other graphlets in the batch serve as negatives. The pre-trained CARTE model is subsequently fine-tuned for downstream tasks."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Language Models for Feature Engineering. Discriminative features enhance the effectiveness of subsequent tabular learning models. Binder [234] identifies task input components that are not directly answerable by a model and leverages LLMs to generate auxiliary features, particularly for knowledge grounding tasks. Given that discriminative features are often manually designed, CAAFE [265] explores the use of LLMs to generate auxiliary features based on task and feature semantics. The quality of these features is then evaluated using a general tabular model, TabPFN [89]. FeatLLM [266] enhances feature generation by incorporating example-based prompting, enabling LLMs to create new features based on textual descriptions. TaPTaP [235] is expected to capture a generic tabular data distribution after ongoing pre-training on a large-scale corpus of real-world tabular data, generating high-quality synthetic tables to support various applications on tabular data."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Language Models for Textual Serialization. A direct approach to incorporating pre-trained language models involves converting tabular data into a textual format, allowing LLMs to infer relationships between features and labels based on embedded expert knowledge. This concept has been validated in semantic parsing tasks [267], [268]. LIFT [236] and TabLLM [80] serialize tabular data by integrating feature names into text and combining them with task descriptions. This enables LLMs to treat tabular prediction tasks as text generation problems. LIFT fine-tunes models on the entire training set, while TabLLM employs few-shot learning for fine-tuning. UniPredict [237] constructs prompts using metadata, sample serialization, and task instructions, fine-tuning LLMs with confidence-weighted augmented labels predicted by an external model. The approach is validated on multiple in-distribution datasets."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Despite their advantages, textual serialization methods face challenges when the number of features increases, as prompts may become too large to fit within the model's context window. The effectiveness of LLMs in tabular data tasks remains constrained by the availability of semantic information and the capabilities of external tabular models. Further exploration of LLM-based methods will be discussed in the general tabular models in Section 7."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "6.4 Reusing a Pre-trained Vision Model"
+ },
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Given the success of deep neural networks (DNNs) in visual tasks, it is intuitive to leverage the strong recognition capabilities of pre-trained vision models for tabular data. Additionally, data augmentation strategies commonly used in image processing can be introduced after transforming tabular data into a visual format. Similar ideas have been explored in time series forecasting [269] and irregular time series classification [270]."
+ }
+ ],
+ [
+ {
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+ "angle": 0,
+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "17"
+ },
+ {
+ "type": "image",
+ "bbox": [
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+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Figure 8: Illustration of transferable tabular methods with a vision model. Tabular data can be transformed into images through dimensionality reduction, table reorganization, and the use of image markers."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The primary challenge lies in representing tabular instances in an image-compatible format. In natural images, neighboring pixels often share semantic relationships, whereas tabular data lacks inherent spatial structure. Features in a tabular instance are permutation-invariant, meaning that exchanging their order does not alter the instance's meaning. Various methods have been proposed to transform tabular data into visual representations, enabling the application of pre-trained vision models fine-tuned for tabular tasks. This subsection highlights different transformation strategies that transfer tabular datasets into images."
+ },
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+ "angle": 0,
+ "content": "Dimensionality Reduction Transformation. Visualization strategies for tabular data naturally convert tables into images by embedding high-dimensional features into a lower-dimensional space. DeepInsight [238] projects tabular data into a 2D space using t-SNE and constructs images through convex hull analysis, applying translation, rotation, quantization, and normalization. REFINED [239] employs Bayesian Metric Multidimensional Scaling to preserve pairwise distances within the low-dimensional representation, ensuring that structurally similar features remain proximate in the transformed image."
+ },
+ {
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+ "content": "Table Reorganization Transformation. A tabular dataset \\(\\mathcal{D}\\) can be treated as a matrix and represented as a single-channel image or kernel. To enable visual PTMs to recognize meaningful spatial relationships, different strategies have been developed for structuring tabular data into images. Tabular Convolution (TAC) [240] arranges data samples into zero-mean square matrices (kernels) of odd integer dimensions. These kernels are then convolved with a fixed \"base image,\" and the resulting images are subsequently fed to a CNN for classification. Image Generator for Tabular Data (IGTD) [74] and TabEye [75] share a similar idea, generating an image for each data sample where pixel intensities correspond directly to feature values. These methods prioritize placing similar features in close proximity but struggle with high-dimensional tabular tasks. LM-IGTD [241] extends IGTD by incorporating stochastic feature generation to enhance robustness and generalization."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Image Marker Transformation. Another approach involves encoding feature values as visual markers within an image. Super-TML [242] assigns feature values to predetermined positions within an image, effectively handling categorical and numerical datasets. Tab2Visual [76] normalizes tabular data and represents each instance as a row of multiple bars, each corresponding to a specific value. Each feature"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "is assigned a unique color to enhance visual differentiation, while bar widths are proportional to feature magnitudes."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "By transforming tabular data into images, these methods enable the application of powerful pre-trained vision models to tabular prediction tasks, leveraging established deep learning techniques from the vision domain to enhance tabular model performance."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "7 FROM TRANSFERABLE TO GENERAL MODEL"
+ },
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The general model (also referred to as the tabular foundation model) represents an advancement over the transferable model. It extends the generalization capabilities of a pretrained tabular model to a variety of heterogeneous downstream tabular tasks, regardless of their diverse feature and class spaces, without requiring additional fine-tuning. In other words, given a pre-trained model \\( g_{\\Theta} \\), it can be directly applied to a downstream tabular task \\( \\mathcal{D} \\) to predict the label of a test instance \\( x^{*} \\) as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ "angle": 0,
+ "content": "\\[\n\\hat {y} ^ {*} = g _ {\\Theta} \\left(\\boldsymbol {x} ^ {*} \\mid \\mathcal {D}\\right). \\tag {7}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Thus, the general model shares similarities with the transferable tabular model, but with a greater emphasis on the \"zero-shot\" ability, aims to construct highly adaptive architectures capable of handling a wide array of heterogeneous datasets simultaneously. Importantly, it does not require an Adapt function, which further reduces the computational cost of hyper-parameter tuning. The goal of the general tabular model is to achieve better generalization on downstream tabular datasets \\(\\mathcal{D}\\) when compared to alternative strategies, such as training a tabular model directly on \\(\\mathcal{D}\\) or adapting a transferable model."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Remark 6. Distinguishing between an advanced transferable tabular model, pre-trained on a wide range of heterogeneous tabular tasks, and the general tabular model can be challenging. Some transferable tabular models, based on auxiliary feature semantics, are able to predict labels for downstream test instances directly [80]. However, their prediction ability is constrained and typically applicable only in specific areas after fine-tuning [78], [233]. The general tabular model, on the other hand, is designed to handle a wider range of heterogeneous tabular tasks, sharing similar pre-training challenges with transferable models but without utilizing additional semantics. Fine-tuning a pre-trained general model is also an option for further performance improvements [93], [96]."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Pre-training has revolutionized domains such as vision and language [271], [84], but its adoption in tabular data remains limited due to the inherent heterogeneity of tabular datasets. Tabular datasets can vary significantly in both dimensionality (i.e., the number of columns) and the semantic meaning of each dimension, even within the same application. For example, different healthcare datasets may capture varying levels of detail and aspects of patient information. Even within the same feature entry (e.g., the \\(d\\)-th column), the meaning can vary (e.g., \"age\" vs. \"height\"). This contrasts with vision and text data (within the same language), where different data sources typically share the same \"vocabulary\" (e.g., pixels, patches, or sub-words) and similar relationships between vocabulary \"elements\" (e.g., neighboring pixels"
+ }
+ ],
+ [
+ {
+ "type": "header",
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+ "angle": 0,
+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "18"
+ },
+ {
+ "type": "image",
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+ "content": null
+ },
+ {
+ "type": "image",
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+ "angle": 0,
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+ },
+ {
+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Figure 9: Illustration of general methods. These methods handle inherent heterogeneity by improving the model's adaptability or homogenizing the diverse tabular formats. Once pre-trained, they can be directly applied to downstream tasks without fine-tuning."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.072,
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+ ],
+ "angle": 0,
+ "content": "often share colors). The lack of shared vocabulary and relationships in tabular data makes it challenging to jointly train a model across multiple datasets, let alone apply a pre-trained model directly to new downstream tasks."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "There are two main strategies to address the inherent heterogeneity in tabular datasets: improving the model's adaptability or homogenizing the diverse tabular formats. We categorize general tabular models into three parts based on their strategies for achieving generalizability. The first focuses on raw-feature-based approaches, among which TabPFN variants represent a rapidly evolving branch and are thus discussed separately. The third category encompasses semantic-based methods that leverage attribute and task semantics to unify heterogeneous tasks."
+ },
+ {
+ "type": "title",
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+ "angle": 0,
+ "content": "7.1 Raw-Feature-based General Models"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "To adapt a general tabular model to heterogeneous tabular datasets during the pre-training and fine-tuning stages, two main strategies can be used from the data-centric and model-centric perspectives. From the data-centric perspective, the general model may standardize tabular datasets into a homogeneous form. For instance, TabPTM [86] transforms all datasets into a uniform format using meta-representation to enable pre-training. The pre-trained model can then be applied directly to a downstream dataset or fine-tuned without introducing additional parameters."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Alternatively, from the model-centric perspective, the general model may improve adaptability by tailoring it to specific tabular tasks. HyperFast [87] adopts the concept of a Hyper Network [272] in meta-learning [273], where a mapping from the tabular dataset to the weights of a classifier is learned. This mapping can then be used to predict labels for unseen instances from the task. To address datasets with varying dimensions, HyperFast projects datasets into a fixed size using random projections. To overcome the slow weight generation speed, MotherNet accelerates HyperFast by modifying its architecture with Transformer-like modules [88]."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "7.2 TabPFN Variants"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The TabPFN family of models [89], [91] leverages the incontext learning capabilities of transformers, directly predicting labels by adapting test instances according to the context of training examples. In the first version of TabPFN, an instance \\( \\boldsymbol{x}_i \\) is padded to a fixed dimension (e.g., 100), and the features are projected to a higher dimension (e.g., \\( d' \\)) for further processing. The label \\( y_i \\) is processed similarly and"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "added to the instance embeddings. The embeddings of all \\( N + 1 \\) instances, including training and test instances, are formulated into a set of \\( N + 1 \\) tokens with \\( d' \\) dimensions. These tokens are processed through several layers of a Transformer, and the output token corresponding to the test instance is further predicted using a 10-way classifier. TabPFN is pretrained over synthetically generated datasets with structured causal models (SCM) [274] and Bayesian Neural Networks (BNNs) [275], [276], enabling the strong in-context learning ability, with the best checkpoint selected based on some real-world datasets. Due to the high complexity of transformers, TabPFN is limited to small-scale tasks, with suggested sizes of \\( N < 1000 \\), \\( d < 100 \\), and \\( C < 10 \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "TabPFN v2 introduces a specialized feature tokenizer to better handle heterogeneity. Specifically, each cell in the table is projected to a \\(k\\)-dimensional vector using a shared mapping, and random position encoding vectors are added to differentiate features [277]. This results in a tensor of size \\((N + 1) \\times (d + 1) \\times k\\) when there is a single test instance. The label of each instance is processed similarly, and the mapped \\(k\\)-dimensional token is concatenated with the instance tokens. A dummy label (e.g., the average of all labels) is used for the test instance since its label is unknown. A two-way attention mechanism is used, with each feature attending to the other features in its row and then attending to the same feature across its column [278]. The output token corresponding to the label of the test instance is further mapped to a 10-class classifier or regressor. Several improvements have been made in TabPFN v2, including increased context size (\\(N < 10000\\), \\(d < 500\\)), automatic feature engineering, and post-hoc ensemble methods. [279] analyzes TabPFN from a bias-variance perspective, shedding light on its generalization capabilities. Various applications have also been explored, including tabular data generation [280], anomaly detection [281], data augmentation [282], and time series forecasting [283]."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The improvements of TabPFN (especially TabPFN v1) stem from several aspects."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Pre-training Improvements. TabForestPFN [284] extends TabPFN by pre-training In-Context Learning (ICL)-transformers on a new forest dataset generator that creates unrealistic datasets with complex decision boundaries. TabDPT [179] pre-trains the architecture on real-world datasets using self-supervised learning and retrieval objectives, making it suitable for both classification and regression tasks. APT [285] is pre-trained utilizing adversarial synthetic data generated by adaptive agents, which systematically modify the underlying data-generating distribution and deliberately challenge the model with diverse synthetic datasets to enhance its robustness and generalization capabilities. TabICL [286] integrates tree-based SCMs using XGBoost [130] to model complex interactions and employs curriculum learning by progressively increasing synthetic dataset sizes. Scalable Improvements. The efficiency of TabPFN is highly sensitive to context size, prompting strategies to enhance scalability and performance [39]. These include compressing training data into a compact learned representation using sketching [287] or prompt tuning techniques [288], [289],"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "1. Some variants of TabPFN are not considered general tabular models, especially the latter parts, as they require additional fine-tuning steps. We place them in this subsection due to their strong relationship with TabPFN."
+ }
+ ],
+ [
+ {
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+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "19"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "employing adaptive data selection methods to identify the most pertinent training examples for each test instance [290], [90], [179], [291], and replacing traditional quadratic attention with computationally efficient linear attention mechanisms [292] and state-space models (SSMs) [293]."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Adaptation Improvements. Some approaches improve TabPFN's performance on downstream tasks by adapting the context [90] or fine-tuning specific parts of the model [96], [284], [290], [289]. TabICL [286] employs a column-then-row attention mechanism to construct fixed-dimensional embeddings of rows, which are subsequently processed by a transformer like TabPFN v1 to facilitate efficient in-context learning. EquiTabPFN [294] introduces self-attention across target components, ensuring that the arbitrary ordering of target dimensions does not influence model predictions, enhancing the performance of TabPFN v1 to some extent."
+ },
+ {
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+ "content": "7.3 Semantics-based General Models"
+ },
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "By leveraging the semantic structure of tabular data, such as column names, heterogeneous tasks can be projected into a shared language space. This allows a single language model, pre-trained on diverse tabular datasets, to handle unseen tasks in a unified manner. TabuLa-8B [92] fine-tunes a Llama 3-8B LLM for tabular data prediction (classification and binned regression) using a novel packing and attention scheme for tabular prediction. GTL [93] transforms tabular datasets into an instruction-oriented language format, facilitating the continued pre-training of LLMs on instruction-oriented tabular data, which demonstrates strong performance in few-shot scenarios. GTL-S [295] unlocks the potential of GTL from a scaling perspective, revealing that scaling datasets and prediction tasks enhance generalization. [94] extends GTL by incorporating retrieval-augmented LLMs for tabular data, combined with retrieval-guided instruction-tuning for LLMs. MediTab [243] uses a data engine that leverages LLMs to consolidate tabular samples to overcome the barrier across tables with distinct schema. MediTab aligns out-domain data with the target task using a \"learn, annotate, and refinement\" pipeline, enabling the pre-trained model to infer for arbitrary tabular input in the domain without fine-tuning."
+ },
+ {
+ "type": "title",
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+ "angle": 0,
+ "content": "8 TABULAR ENSEMBLE METHODS"
+ },
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+ "type": "text",
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+ "content": "Ensemble learning is a natural way to improve the generalization ability of multiple base learners by leveraging their diversity. Classical methods such as Random Forest [127] and AdaBoost [126], [296] employ bagging and boosting, respectively, by ensembling multiple decision trees. These methods have proven effective for tabular data, as they reduce bias/variance and improve robustness [297]."
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+ "content": "In deep tabular learning, ensemble methods can be categorized into two primary approaches: joint-training ensembles, where multiple sub-networks are aggregated within a single training pipeline, and post-hoc ensembles, where the predictions from multiple pre-trained deep tabular models are fused. One major challenge in ensembling deep tabular methods is computational efficiency, as training multiple deep models or sub-models can be computationally expensive and time-consuming."
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+ "content": "8.1 Joint-Training Ensembles"
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+ "content": "Joint-training ensemble methods integrate diverse model architectures within a single training process to improve predictive performance while maintaining efficiency. These architectures often combine different types of models, such as linear and non-linear models [28] or tree-based and deep neural network-based approaches [63]. Tree-mimic methods leverage this concept by mixing predictions from multiple tree nodes to enhance robustness [60], [59], [193]."
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+ "content": "To improve efficiency while maintaining predictive power, various techniques have been explored. Some approaches employ parameter-efficient ensembles, such as TabM [176], which uses MLPs as base learners and incorporates BatchEnsemble [298] to generate multiple diverse base learners efficiently. This prevents a large increase in the number of learnable parameters while maintaining model diversity. Similarly, BETA leverages pre-trained TabPFN by generating multiple base learners through additional parameter tuning [96]. Specifically, BETA learns multiple feature projections, feeding the projected training sets into TabPFN and aggregating the results while applying BatchEnsemble to reduce the number of additional learnable parameters."
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+ "content": "Some hybrid approaches, such as LLM-Boost and PFN-Boost, have been developed to integrate large language models and TabPFN with gradient-boosted decision trees [299]. In these approaches, LLMs and PFN serve as the initial base learners, and additional base learners are sequentially trained in a boosting manner. This approach leverages the strong prior knowledge from LLMs and TabPFN while maintaining the scalability of gradient-boosted decision trees."
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+ "content": "8.2 Post-Hoc Ensembles"
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+ "content": "Post-hoc ensemble (PHE) methods involve combining multiple trained models to improve robustness and accuracy. Bagging-based ensembles are one of the most direct post-hoc strategies, where usually multiple models trained with different random seeds are aggregated [33], [69]. Although this approach improves model robustness, it incurs high computational overhead. Some recent studies have demonstrated that LLM-based methods exhibit diverse prediction behaviors compared to deep tabular models that do not utilize attribute names [94]. This difference in prediction styles enhances their complementarity, making them ideal candidates for ensemble methods."
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+ "content": "Instead of explicitly training multiple models, perturbation-based approaches create diverse predictions from the same pre-trained model. One such method applies feature permutation with TabPFN, leveraging the fact that TabPFN is not fully feature permutation-invariant [89]. A perturbation-based ensemble can be formed by randomly permuting the feature order in both the training and test sets and making predictions multiple times, generating multiple diverse predictors without additional training costs. TabPFN v2 introduces additional perturbations to enhance diversity among several key factors, including variations in feature encoding, feature quantization, categorical feature shuffling, SVD-based feature compression, outlier removal, and power transformations such as the Yeo-Johnson transformation [91]. These randomly selected transformations create diverse"
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+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
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+ "content": "prediction patterns, enabling effective ensemble learning without requiring multiple separately trained models."
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+ "content": "Another post-hoc ensemble strategy employed in TabPFN v2 is the use of Portfolio-Based Ensemble, where a fixed set of TabPFN configurations is used [91]. A greedy ensemble selection technique is then applied to learn optimal weights for aggregating the predictions of different configurations [300]. By combining multiple perturbed models, this method improves generalization without excessive training costs. Some methods apply ensemble techniques to TabPFN v1 to handle large datasets. For instance, TabPFN-Bagging [96], [301] divides large datasets into multiple context groups, with the final results averaged to mitigate variance. BoostPFN [301] treats TabPFN v1 as weak learners, where each weak learner uses a subset of the training data as context. This approach allows BoostPFN to outperform standard Prior Fitted Networks (PFNs) on large datasets."
+ },
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+ "content": "9 EXTENSIONS"
+ },
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+ "content": "In this section, we briefly introduce some extensions on deep tabular methods across different complex tasks."
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+ "content": "Clustering. Traditional clustering approaches often leverage enhanced distance metrics, such as the Gower distance [302], which is specifically designed for mixed data types, and interpretable prototypes, such as K-medoids. Recent advances in tabular data clustering have sought to integrate interpretability constraints with deep representation learning. For example, IDC [97] introduces a deep learning framework for general tabular data that predicts interpretable cluster assignments at both the instance and cluster levels. To address overlapping clusters, TableDC [98] integrates the Mahalanobis distance, which accounts for variance and correlation within the data. This method provides a similarity measure suitable for tables, rows, or columns in high-dimensional latent spaces."
+ },
+ {
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+ "content": "Anomaly Detection. Anomaly detection in tabular data is crucial for identifying subtle irregularities in structured datasets, such as fraudulent transactions or equipment failures. While classical techniques like Isolation Forest [303] and Local Outlier Factor [304] remain foundational, recent developments have incorporated various methods to capture contextual relationships in high-dimensional data. For instance, [305] introduces a method that learns mappings that maximize mutual information between each sample and the part that is masked out, capturing the structural nuances of samples from a single training class. ADBench [99] provides a comprehensive tabular anomaly detection benchmark with 30 algorithms and 57 benchmark datasets. Additionally, large language models (LLMs) have also been employed for anomaly detection in tabular data [306]."
+ },
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+ "content": "Tabular Generation. Tabular data generation has become an essential tool for synthetic data creation, privacy preservation, and addressing data scarcity. Traditional methods, such as Bayesian networks or GANs, focus on mimicking marginal distributions, while recent advancements emphasize preserving complex feature dependencies and semantic consistency. For instance, tabular diffusion models [307] iteratively refine synthetic data to capture subtle correlations in high-dimensional datasets, outperforming GANs in terms of data"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "fidelity. [308] introduces high-order structural causal information as a natural prior knowledge and offers a benchmark framework for evaluating tabular synthesis models. Despite these advances, challenges remain in balancing realism with privacy, such as avoiding identity leakage in sensitive datasets, and scaling to heterogeneous data types. Hybrid neuro-symbolic models [309] bridge this gap to provide trustworthy synthetic data for downstream tasks."
+ },
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+ "angle": 0,
+ "content": "Interpretability. Traditional gradient-boosted decision trees (GBDTs) inherently provide interpretability through feature importance scores and decision path visualization. Frameworks such as XGBoost [130] and LightGBM [131] quantify feature importance using metrics like split frequency and information gain. SHAP values [310] enable instance-level explanations by decomposing model predictions into feature contributions. The additive nature of GBDTs allows for partial dependence plots [311] to visualize feature effects while controlling for interactions. NeC4.5 [253], a novel decision tree algorithm that integrates the comprehensibility of decision trees with the generalization ability of neural network ensembles. By training a neural network ensemble to generate a new training set, NeC4.5 enhances decision tree performance while maintaining interpretability."
+ },
+ {
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+ "bbox": [
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+ "angle": 0,
+ "content": "Recent deep models specifically designed for tabular data have introduced novel interpretability mechanisms. For example, NAMs [255] combine some of the expressivity of DNNs with the inherent intelligibility of generalized additive models. They learn a linear combination of neural networks that each attend to a single input feature, which are trained jointly and can learn arbitrarily complex relationships between their input feature and the output. TabNet [105] uses sequential attention with learnable feature masks, where each decision step explicitly selects a subset of features via sparse masking. The aggregated feature usage across steps provides global interpretability comparable to GBDT's feature importance. Subsequent variants, such as TabTransformer [63], enhance interpretability by visualizing cross-feature attention patterns. FT-Transformer [33] combines feature tokenization with explainable attention, while NODE [60], NODE-GAM [61] and DOFEN [312] generalize ensembles of oblivious decision trees, benefiting from both end-to-end gradient-based optimization and multi-layer hierarchical representation learning."
+ },
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+ "bbox": [
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+ "content": "Open-Environment Tabular Machine Learning. Research on distribution shifts in tabular data starts with domain-to-domain shifts [110], which are commonly categorized based on the availability of target domain data. When target data is available, transfer learning techniques such as unsupervised domain adaptation [313] and test-time adaptation [314] are widely used. These methods adapt model parameters using test-time inputs but rely on access to target distributions, which may not always be feasible. In contrast, when target data is unavailable, a more practical but challenging scenario, methods aiming to enhance robustness and generalization, using approaches such as domain generalization [315], domain robustness [316], [317], label robustness [318] or ensemble strategies [95]. TableShift [110] provides a detailed analysis of this scenario."
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+ "angle": 0,
+ "content": "Beyond domain-to-domain shifts, temporal shifts are more general and complex. TabReD [109] emphasizes the inherent temporality of real-world tabular data, advocating"
+ }
+ ],
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+ "angle": 0,
+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "content": "21"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "for temporal splits for training and testing. [319] further propose a refined training protocol focusing on temporal evaluation, significantly improving generalization across models. To address temporal shifts, it's critical to incorporate temporal information [319]. Drift-Resilient TabPFN [174] models temporal shifts with a secondary SCM, which specifies changes in the primary model parameters. [319] introduce a plug-and-play temporal embedding that effectively captures trend and periodicity patterns, providing an adaptive mechanism to mitigate the impact of temporal shifts. Under temporal shift conditions, most methods experience performance degradation, while TabM [95] exhibits relative robustness [109]. However, [319] demonstrate that with the refined training protocol and temporal embedding, methods such as ModernNCA [35] can regain competitiveness."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Multi-modal Learning with Tabular Data. Text, such as feature names, can be effectively utilized to enhance tabular data learning, as discussed in Section 6. Here, we focus on interactions with the image modality, e.g., in healthcare, where medical images require specialized equipment and expert knowledge, often in tabular form, for accurate diagnosis [320]. To tackle challenges like large medical datasets and high annotation costs, MMCL [106] uses a contrastive self-supervised learning framework that integrates images and tabular data. CHARMS [107] transfers expert knowledge from tabular data to images, improving image predictions even without tabular data during inference, thus reducing reliance on costly expert annotations. TIP [321] proposes a self-supervised learning strategy with a tabular encoder for incomplete, heterogeneous data and a multimodal interaction module for inter-modality representation learning."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Tabular Understanding. Tabular understanding involves comprehending the information contained within a table and can be broken down into several tasks. For example, Table Detection (TD) [322], [323] refers to identifying the region of the image that contains the table while Table Structure Recognition (TSR) [324], [325] involves the identification of the rows and columns to identify individual table cells, which aims to recognize the cellular structures of tables from table images by extracting the coordinates of cell boxes and row/column spanning information. Table Question Answering (TQA) [326], [327], [112] refers to providing precise answers from tables to answer a user's question. Traditional methods, whether OCR-based [328], [329], [330] or OCR-free [331], [332], [333], [334], [335], have made significant strides in TSR and TD, which are relatively simpler tasks."
+ },
+ {
+ "type": "text",
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+ "content": "More complex tasks, such as TQA, have also been the focus of considerable effort. For example, Donut [332] proposes a novel task and a synthetic document image generator to pre-train the model, reducing reliance on large-scale real document images. Monkey and TextMonkey [336], [337] utilize shifted window attention and use similarity measures to filter out redundant tokens. mPLUG-DocOwl [338] adapts mPLUG-Owl for OCR-free document understanding, while TabPedia [335] constructs low- and high-resolution vision encoders with a concept synergy mechanism for visual table understanding. [339] focuses on exploring various table representations and directly prompting LLMs to improve performance. Please refer to [112], [113] for more details."
+ },
+ {
+ "type": "title",
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+ "angle": 0,
+ "content": "10 DISCUSSIONS"
+ },
+ {
+ "type": "text",
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+ "angle": 0,
+ "content": "In this section, we discuss several possible future directions for tabular machine learning, particularly in light of the significant potential demonstrated by tabular general/foundation models."
+ },
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The Ability to Handle Dynamic and Open Environments. Tabular models, particularly foundation models, will increasingly need to operate in dynamic, real-world environments where data evolves over time [340]. One of the key challenges is dealing with imbalanced datasets [155], where certain classes may be underrepresented, and the distribution of data may shift over time [110]. As a result, models need to adapt to these changes and continue providing accurate predictions. Additionally, the emergence of new classes in the data may require the model to evolve and update its predictions in real-time [341]. This calls for methods that ensure tabular foundation models can accommodate evolving data, handling both new classes and changing distributions effectively."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The Coverage and Scope of Tabular Foundation Models. Current tabular foundation models have demonstrated strong performance on various unseen classification and regression tasks. However, several important questions remain about their capabilities. For instance, in addition to in-context learning [246], are there other prediction strategies that could be employed to further enhance the versatility and performance of tabular foundation models? Beyond classification and regression, can these models be extended to handle related tasks such as clustering, imputation, outlier detection, or even table-based question answering (QA)? Expanding the task scope could increase the model's utility in a wide range of applications. Furthermore, it is worth investigating whether there is a scaling law [342] for tabular foundation models. Currently, tabular checkpoints are relatively small compared to foundation models in other modalities, such as language or vision. Understanding the implications of scaling these models—particularly the trade-offs between model size and performance—will be crucial for their future development."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Will Foundation Models Always Help? While foundation models have demonstrated impressive generalization abilities, there are inherent trade-offs. Similar to ensemble learning, a single foundation model may provide an \"average\" predictive ability across tasks, potentially losing specialized expertise for specific tasks. To address this, a promising approach could be the development of a \"tabular model zoo\" [343], [344]. In this paradigm, different pre-trained models, potentially including models from other domains, could be combined for a specific tabular task. Given a new task, suitable pre-trained models could be selected, adapted if necessary, and integrated for improved performance."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Model Efficiency. In many real-world applications, tabular datasets are large and high-dimensional, posing significant challenges for both training and inference [345], [44]. One area of concern is how to handle extreme cases, such as when the data is exceptionally large or sparse. Foundation models should be able to scale effectively in these scenarios without sacrificing performance. Another issue is inference speed. In large-scale problems, timely predictions are essential, especially when deployed in real-time environments [292]. Opti-"
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+ ],
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+ "bbox": [
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+ "angle": 0,
+ "content": "JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, XXXX 20XX"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "22"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "mizing the inference process is therefore critical to ensure that predictions can be made quickly on large, complex datasets. Lastly, the computational resources required for training and deploying foundation models can be substantial [346]. Optimizing resource usage through methods such as model pruning, quantization, and efficient training algorithms will be important to ensure that these models remain practical and accessible for a wide range of applications."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Bridging the Gap Between Tabular Data and Other Modalities. Tabular data often coexists with other data modalities, such as images and text. One of the exciting challenges in the field is how to effectively integrate tabular data with foundation models from other domains [347]. Combining the strengths of tabular models with those of vision or language models could result in more powerful and versatile models capable of handling multimodal data. Exploring how to seamlessly integrate these modalities—whether through joint embeddings, cross-modal attention mechanisms, or other techniques—could unlock significant advances in tasks that require both structured tabular data and unstructured data sources like images or text."
+ },
+ {
+ "type": "title",
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+ "angle": 0,
+ "content": "11 CONCLUSION"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Tabular data remains a cornerstone of real-world machine learning applications, and the advancement of deep learning has opened new possibilities for effective representation learning in this domain. In this survey, we present a comprehensive overview of deep tabular representation learning, covering its background, challenges, evaluation benchmarks, and the discussion between tree-based models and DNNs. We systematically categorize existing methods into three categories—specialized, transferable, and general models—based on their generalization capabilities. In addition, we discuss ensemble techniques, extensions, and some promising future directions, such as open-environment and multimodal tabular learning. We hope this survey serves as a valuable reference for understanding the current state of the field and inspires further progress in developing more robust and generalizable tabular learning methods."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "REFERENCES"
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+# Representation Learning for Tabular Data: A Comprehensive Survey
+
+Jun-Peng Jiang, Si-Yang Liu, Hao-Run Cai, Qile Zhou, Han-Jia Ye
+
+Abstract—Tabular data, structured as rows and columns, is among the most prevalent data types in machine learning classification and regression applications. Models for learning from tabular data have continuously evolved, with Deep Neural Networks (DNNs) recently demonstrating promising results through their capability of representation learning. In this survey, we systematically introduce the field of tabular representation learning, covering the background, challenges, and benchmarks, along with the pros and cons of using DNNs. We organize existing methods into three main categories according to their generalization capabilities: specialized, transferable, and general models. Specialized models focus on tasks where training and evaluation occur within the same data distribution. We introduce a hierarchical taxonomy for specialized models based on the key aspects of tabular data—features, samples, and objectives—and delve into detailed strategies for obtaining high-quality feature- and sample-level representations. Transferable models are pre-trained on one or more datasets and subsequently fine-tuned on downstream tasks, leveraging knowledge acquired from homogeneous or heterogeneous sources, or even cross-modalities such as vision and language. General models, also known as tabular foundation models, extend this concept further, allowing direct application to downstream tasks without additional fine-tuning. We group these general models based on the strategies used to adapt across heterogeneous datasets. Additionally, we explore ensemble methods, which integrate the strengths of multiple tabular models. Finally, we discuss representative extensions of tabular learning, including open-environment tabular machine learning, multimodal learning with tabular data, and tabular understanding tasks. More information can be found in the following repository: https://github.com/LAMDA-Tabular/Tabular-Survey.
+
+Index Terms—Tabular Data, Representation Learning, Deep Tabular Learning, Tabular Machine Learning, Tabular Foundation Model
+
+
+
+# 1 INTRODUCTION
+
+Tabular data, characterized by structured rows and columns, is one of the most prevalent data formats in real-world machine learning applications, spanning diverse domains such as finance [1], healthcare [2], education [3], recommendation systems [4], and scientific research. In particular, AI for scientific research (AI4science) has increasingly relied on tabular data, as numerous prominent datasets—such as those from genomics [5], chemistry [6], and climate science [7], [8]—naturally adopt tabular forms.
+
+Tabular data inherently organizes information in a structured, table-like format. In this survey, we focus primarily on supervised tabular machine learning tasks, specifically classification and regression. Beyond their structured organization, tabular datasets frequently include heterogeneous attributes [9], encompassing numerical, categorical, or mixed data types that may be either dense or sparse. Additionally, many tabular datasets present quality challenges, such as noisy measurements, missing values, outliers, inaccuracies [10], and privacy constraints [11], all of which complicate the modeling process. The most common supervised tabular tasks are classification and regression, where the goal is to learn mappings from training data to discrete or continuous targets, respectively. As illustrated in Figure 1, each row represents an instance (with its corresponding label), while each column corresponds to a specific attribute or feature [12].
+
+
+Figure 1: A brief introduction to tabular data and associated learning tasks. Each row represents an instance and each column corresponds to a specific attribute or feature, which can be numerical or categorical. The most common tabular machine learning tasks are classification and regression as shown in the right side of the figure.
+
+Ideally, learned mappings should generalize effectively, accurately predicting outcomes for new instances drawn from the same underlying distribution.
+
+Machine learning methods for tabular data have evolved significantly over the years [13], [14], [15], [16]. Recently, the rise of deep learning has profoundly impacted domains like computer vision [17] and natural language processing [18], where Deep Neural Networks (DNNs) extract semantic representations directly from raw inputs [19], [20], [21]. These learned representations have not only improved generalization but have also facilitated knowledge transfer across related tasks [22]. The flexibility of DNNs in modeling complex feature interactions and learning rich hierarchical
+
+
+Figure 2: We organize existing tabular classification/regression methods into three categories according to their generalization capabilities: specialized (left), transferable (middle), and general (right) models. Specialized models focus on tasks where training and evaluation occur within the same data distribution. Transferable models are pre-trained on one or more datasets and subsequently fine-tuned on downstream tasks. General models, also known as tabular foundation models, extend this concept further, allowing direct application to downstream tasks without additional fine-tuning.
+
+structures has inspired significant interest in adapting deep learning techniques to tabular data.
+
+Indeed, DNNs were applied to tabular data decades ago, initially targeting dimensionality reduction and visualization tasks [23], [24], [25], [26], yet they typically struggled to match tree-based methods on standard classification and regression problems. Later advances in DNNs have led to significant improvements across various tabular-related applications, such as click-through rate prediction [27], [28], anomaly detection [29], recommendation systems [30], and time series forecasting [31], [32]. Modern deep learning approaches, benefiting from better-designed architectures, optimized training strategies, high-quality representations, have revitalized DNN performance on tabular data, often rivaling or surpassing traditional tree-based models [33], [34], [35]. Given the wide variety of approaches emerging in deep tabular modeling, a systematic overview that revisits critical factors and current methodologies in representation learning for tabular data has become increasingly necessary.
+
+This survey begins by introducing the background of tabular data learning, highlighting the challenges involved and critically examining the advantages and limitations of utilizing DNNs compared to classical—particularly tree-based—methods [36], [37], [38], [39]. Given the observed instability of method performance across different tabular datasets, we also discuss comprehensive strategies for dataset collection, evaluation, and analysis, aiming to establish robust criteria for aggregating performance metrics across multiple datasets [40], [41], [42], [43].
+
+We broadly categorize deep tabular methods into three types: specialized methods, transferable methods, and general methods, distinguished by the scope of datasets on which they are trained and deployed, as well as their corresponding
+
+generalization capabilities (illustrated in Figure 2). Specialized tabular methods align closely with classical supervised models, typically trained and evaluated on data drawn from the same distribution. In contrast, transferable methods leverage knowledge from models pre-trained on one or multiple source datasets, subsequently fine-tuning these models on target datasets; the primary challenge here lies in addressing the heterogeneity between pre-trained sources and target tasks. The recently proposed general tabular methods—motivated by the remarkable "zero-shot" generalization abilities demonstrated by large language models (LLMs)—exhibit exceptional versatility. These general models can directly apply their learned representations to downstream tabular datasets without additional fine-tuning, achieving robust generalization due to advanced pre-training strategies. Although the generalization ability tends to increase from specialized to general models, it does not imply that specialized or transferable methods are less valuable; specialized models remain superior on large-scale datasets, and fine-tuning general models can further improve their predictive performance. Additionally, the first two types of methods provide foundational insights and valuable components that contribute significantly to advancements in general tabular models.
+
+For specialized methods, numerous designs have been proposed from diverse perspectives, and previous papers have often categorized these methods based primarily on their architectural characteristics or behaviors. Existing taxonomies [44], for example, group specialized methods into feature-preprocessing-based [33], [45], data-augmentation-based [46], [47], [48], [49], MLP variants [50], [34], specialized DNN architectures [51], [52], [53], [54], [55], [56], [57], [58], tree-mimic approaches [59], [60], [61], token-based tech
+
+niques [62], [63], [33], [64], [65], regularization-driven methods [66], [67], and neighborhood-based strategies [68], [69], [35]. However, such categorizations can appear scattered, making it difficult to connect the core ideas between methods placed in distinct groups. In contrast, this survey introduces a hierarchical taxonomy based on the key aspects of tabular data—features, samples, and objectives—providing a cohesive organizational framework. Our approach emphasizes detailed strategies for obtaining high-quality representations at both feature- and sample-levels. This unified perspective helps bridge core ideas across diverse methods, facilitating clearer comparative discussions and potentially guiding the design of future, more advanced tabular models.
+
+Instead of training a model from scratch on a single tabular dataset, transferable models leverage knowledge encoded in a pre-trained model from another dataset, which can significantly enhance the training process, especially when data or computational resources for the target task are limited. A major challenge in transferring knowledge across tabular tasks lies in the inherent heterogeneity between the source and target datasets, particularly differences in their feature and label spaces. In this survey, we adopt a broad perspective on transferable tabular models, categorizing methods based on the sources of their pre-trained knowledge. Specifically, we discuss models pre-trained on homogeneous tabular domains, such as self-supervised methods with additional pre-training steps on the target dataset itself [70], [71]; models pre-trained across heterogeneous tabular domains [72], [73], [64]; and methods transferring knowledge from other modalities, such as vision-based pre-trained models [74], [75], [76]. Additionally, since incorporating attribute semantics (when available) is a common strategy for bridging heterogeneous attribute spaces across tabular datasets [77], [78], [79], we also explore approaches leveraging language models in the final category. In particular, we further organize these language model-based strategies according to the methods they use to extract knowledge and the types of language models involved—ranging from small-scale language models to Large Language Models (LLMs) [80], [81], [82], [83].
+
+Inspired by recent advancements in foundation models from vision and language domains [84], [85], general models—also known as tabular foundation models—expand the concept of transferable tabular models by enabling direct application to downstream tasks without additional fine-tuning. This capability, commonly referred to as the model's "zero-shot" ability, significantly enhances the model's usability across diverse tabular datasets. In contrast to transferable models, which primarily focus on bridging knowledge gaps between source and target datasets, general models aim to construct highly adaptive architectures capable of handling a wide array of heterogeneous datasets simultaneously. We categorize these general models based on the strategies used to achieve adaptiveness across diverse tabular tasks, specifically examining adaptations from both data-centric [86] and model-centric perspectives [87], [88]. Furthermore, we discuss critical branches of general tabular models in detail: the TabPFN variants leveraging in-context learning [89], [90], [91], and methods utilizing attribute and task semantics to unify heterogeneous tasks within a common representation framework [92], [93], [94].
+
+Additionally, ensemble methods [95], [96], [91] are in
+
+produced, which improve the generalization ability based on the strengths of multiple tabular models. Finally, we briefly overview other relevant extensions of tabular learning, including clustering [97], [98], anomaly detection [99], [100], [101], data generation and imputation [102], [103], [104], interpretability [63], [105], [61], multimodal learning [106], [107], open-environment tabular machine learning [108], [109], [110], [111], and tabular understanding [112], [113]. By summarizing the state of the field and identifying open challenges, we aim to guide future research and applications in tabular data representation learning.
+
+# 2 BACKGROUND
+
+This section presents the (supervised) tabular machine learning task, including the notation of tabular data learning, the history of tabular data, the challenges of learning from tabular data, evaluation metrics, and tabular benchmarks.
+
+# 2.1 Learning with Tabular Data
+
+A supervised tabular dataset is formatted as $N$ examples and $d$ features/attributes corresponding to $N$ rows and $d$ columns in the table. An instance $\pmb{x}_i\in \mathbb{R}^d$ is depicted by its $d$ feature values. Assume $x_{i,j}$ as the $j$ -th feature of instance $\pmb{x}_i$ , it could be a numerical (continuous) one $x_{i,j}^{\mathrm{num}}\in \mathbb{R}$ , like the temperature of a region or the density of the object. $\pmb{x}_i$ can also be a categorical (discrete) value $x_{i,j}^{\mathrm{cat}}$ , like one of multiple colors, the location of a person, or even some textual descriptions of the instance. Each instance is associated with a label $y_i$ , where $y_i\in \{1, - 1\}$ in a binary classification task, $y_i\in [C] = \{1,\dots ,C\}$ in a multi-class classification task, and $y_i\in \mathbb{R}$ in a regression task.
+
+Remark 1. Ordinal regression [114], [115], also called ordinal classification, is a type of regression analysis used to predict an ordinal variable. It can be considered an intermediate problem between regression and classification. However, this survey primarily focuses on standard classification and regression tasks and does not specifically discuss ordinal regression.
+
+Given a tabular dataset $\mathcal{D} = \{(x_i, y_i)\}_{i=1}^N$ , we aim to learn a mapping $f$ on $\mathcal{D}$ that maps $x_i$ to its label $y_i$ . In other words, the model predicts $x_i$ with $\hat{y}_i = f(x_i)$ . The general objective learning $f$ follows the structural risk minimization:
+
+$$
+\min _ {f} \sum_ {\left(\boldsymbol {x} _ {i}, y _ {i}\right) \in \mathcal {D}} \ell (y, \hat {y} _ {i} = f \left(\boldsymbol {x} _ {i}\right)) + \Omega (f). \tag {1}
+$$
+
+$\ell (\cdot ,\cdot)$ measures the discrepancy between the predicted label $\hat{y}_i$ and the true label $y_{i},e.g.$ , cross-entropy in classification and mean square error in regression. $\Omega (\cdot)$ is the regularization on the model, which restricts the complexity of $f$ . We expect the learned $f$ is able to extend its ability to unseen instances sampled from the same distribution as $\mathcal{D}$ .
+
+Tabular methods differ in their strategies to implement $f$ . The "dummy" approach makes predictions based on training labels $\{y_i\}_{i=1}^N$ directly, which outputs the major class in the training set for classification and the average of all labels for regression, respectively.
+
+In a $C$ -class classification task, classical parametric methods implement $f$ with a linear mapping, i.e., $f(\pmb{x}_i) = \pmb{W}^\top \pmb{x}_i + \pmb{b}$ , where the classifier $\pmb{W} \in \mathbb{R}^{d \times C}$ and $\pmb{b} \in \mathbb{R}^C$
+
+is the bias. With different loss functions, we can implement Logistic Regression, SVM, or even AdaBoost. In contrast, non-parametric methods implement the prediction via $f(\pmb{x}_i) = f(\pmb{x}_i, \mathcal{D})$ , depending on the whole training set. For example, KNN searches neighbors in the training set $\mathcal{D}$ with the $K$ smallest distance w.r.t. $\pmb{x}_i$ . KNN can be viewed as a specific label smoother, with a dynamic local region for every instance. [116] links KNN and Random Forest from their ways of smoothing training labels in their predictions.
+
+Deep tabular methods implement $f$ with a deep neural network, e.g. Most deep learning models could be decomposed into two parts, i.e., $f(\pmb{x}_i) = \pmb{W}^\top \phi(\pmb{x}_i) + \pmb{b}$ . Similar to the linear model, $\pmb{W}$ and $\pmb{b}$ are the components of linear classifier, with $\pmb{W} \in \mathbb{R}^{d' \times C}$ . $\phi$ maps the input vector $\pmb{x}_i$ into the $d'$ dimension space, which extracts semantic embeddings for the given tabular input. $\phi$ could be implemented with MLP or residual network.
+
+# 2.2 History of Tabular Data
+
+Historically, classical machine learning tasks were predominantly formulated with tabular data, or datasets readily transformed into a tabular representation without explicitly designating them as "tabular." In early literature, the term "tabular" typically referred to tables within relational databases [117], CSV files on the web [118], or tables in documents [119]. Relevant tasks included table extraction [120], parsing [121], understanding [122], and discovering association rules [123]. With the expansion of machine learning applications into other modalities such as images, texts, audio, and video, the classical vector-based data representations have come to be explicitly termed "tabular data."
+
+Early statistical approaches such as linear regression, logistic regression, linear discriminant analysis, and K-Nearest Neighbors (KNN) predate artificial intelligence. Classical learning methods further expanded across various paradigms, including decision trees [124], [125], multi-layer perceptrons (MLPs), support vector machines (SVMs), and nearest centroid classifiers [5], [14]. Ensemble methods enhanced predictive performance by aggregating outputs from multiple base learners [126], [127]. More recently, gradient boosting frameworks [128], [129], such as XGBoost [130], LightGBM [131], and CatBoost [132], have become prominent due to their effectiveness and efficiency in tabular data applications and competitions [133], [134].
+
+With the development of deep learning, DNNs were applied to tabular classification and regression tasks decades ago, utilizing architectures such as stacked Restricted Boltzmann Machines and denoising autoencoders [135], [136], [137]. Early representation learning efforts primarily focused on dimensionality reduction and data visualization tasks [23], [24], [25], [26], yet these models struggled to surpass traditional tree-based methods in terms of generalization. However, advancements in neural network architectures and representation learning strategies have recently led to promising results in related tabular domains, including click-through rate prediction [27], [28], anomaly detection [138], [29], recommendation systems [139], [30], and time series forecasting [31], [140], [32], [141]. Innovations such as convolutional layers and learnable feature embeddings have improved the ability of deep models to capture high-order
+
+attribute relationships [142], [143]. While early deep tabular methods lagged behind ensemble tree-based models, recent techniques have demonstrated competitive or superior performance [33], [34], [35], affirming deep representation learning as a promising direction for tabular data modeling.
+
+While several survey papers have been published [9], [144], the field of tabular data has witnessed remarkable progress over the past two years. On one hand, the emergence of new specialized methods has introduced significant shifts in the landscape, motivating the need for our comprehensive taxonomy. On the other hand, the rise of transferable and general approaches has greatly enhanced the generality and applicability of tabular data modeling, which has been overlooked in previous works.
+
+# 2.3 Challenges of Learning from Tabular Data
+
+Different from other types of data sources, e.g., images and texts, there exist several challenges dealing with tabular datasets due to their characteristics.
+
+Heterogeneity of Features. Unlike continuous image data or token-based textual data, tabular datasets often contain both numerical and categorical attributes, each requiring distinct handling methods [9], [145]. Numerical features frequently exhibit varying ranges and distributions, necessitating normalization or scaling. Categorical features differ in cardinality and semantic interpretation, requiring encoding methods like one-hot vectors or embeddings. Consequently, tabular models must carefully handle these mixed data types to preserve the usability of each feature.
+
+Lack of Spatial Relationships. Tabular data inherently lacks spatial or sequential relationships that are naturally found in other modalities [74], [50]. The order of columns has no semantic or spatial meaning, making tabular data permutation-invariant regarding features. Moreover, standard tabular machine learning assumes rows are independently and identically distributed (i.i.d.), further eliminating temporal or sequential correlations present in data such as video or time series. This absence of inherent spatial or sequential structure challenges deep learning architectures traditionally designed to exploit such dependencies.
+
+Low-quality and Missing Data. Compared to image or text data, where contextual or spatial redundancies help manage missing or corrupted values, tabular data is more vulnerable to incomplete or erroneous entries [146], [147]. Missing values in tabular datasets can introduce significant biases and degrade prediction quality. Additionally, noisy or incorrect values can considerably affect model reliability. Data preprocessing steps, including data cleaning and imputation, become crucial to maintaining accuracy and robustness in tabular machine learning.
+
+Importance of Feature Engineering. Effective tabular models heavily depend on the quality of their input features [45], [148]. Unlike image or textual data, where DNNs inherently learn feature representations from raw data, tabular methods often require domain-specific knowledge and meticulous manual feature engineering. Identifying and modeling complex, nonlinear interactions among tabular features frequently demands sophisticated transformations and expert insight, significantly impacting the predictive performance of models [149].
+
+Class Imbalance. Tabular datasets frequently exhibit imbalanced label distributions, especially in classification tasks, where certain categories are underrepresented [150], [151]. Class imbalance complicates model learning, leading to biased outcomes toward majority classes and poor performance on minority classes. Specialized methods such as oversampling, undersampling, or tailored loss functions (e.g., focal loss [152]) are required to address this imbalance effectively. Evaluation criteria like the AUC or F1-score further help assess model quality in imbalanced settings. Recent research highlights differences between deep and classical models in handling imbalance, emphasizing the need for careful consideration [153], [154], [155], [41].
+
+Remark 2. Class imbalance has long been a known issue in the tabular domain, even before the rise of deep learning [156], and methods such as SMOTE [157], [158] can easily be extended to deep learning methods during preprocessing. However, Current deep tabular methods primarily assume that the training and testing data come from the same distribution, even in cases involving class imbalance. In addition, some class imbalance methods in visual domain can be readily extended to the tabular data learning [159], [160]. Therefore, we do not delve into class imbalance in this survey.
+
+Scalability to Large Datasets. Tabular datasets can become large-scale and high-dimensional, presenting computational and generalization challenges [161]. With increasing dimensionality, the risk of overfitting increases, especially when the number of features significantly surpasses the number of samples. Consequently, efficient training algorithms, memory management strategies, and sufficient computational resources become essential. Effectively scaling tabular models to handle large datasets while maintaining generalization ability remains a challenging but critical research area [162]. Model Selection and Hyperparameter Tuning. Tabular models are particularly sensitive to hyperparameter settings [163], [164]. Selecting an appropriate model architecture and tuning hyperparameters, such as learning rate, layer depth, or number of trees, can be computationally expensive and time-consuming. Despite the advancement of automated machine learning (AutoML) techniques [165], [166], [167], efficiently identifying optimal configurations for deep tabular methods under practical constraints remains challenging and critical for achieving high predictive performance.
+
+Domain-Specific Constraints. Certain application domains, such as healthcare or finance, impose additional regulatory or ethical requirements on model development [168]. For example, healthcare applications must comply with privacy standards like HIPAA [169] and provide explainability to clinicians. Financial models similarly must adhere to fairness regulations and industry standards. These constraints can restrict algorithm selection, necessitate interpretable predictions, and require additional validation, explainability, and auditability procedures [170], [171], [172].
+
+# 2.4 Evaluation of a Tabular Method
+
+We present the evaluation of tabular methods, ranging from traditional to modern, to provide a comprehensive evaluation across different aspects. For a given model on a dataset $\mathcal{D}$ ,
+
+we employ standard metrics that quantify the discrepancy between the predicted label $\hat{y}_i$ and the true label $y_i$ .
+
+Evaluation on A Single Task. For classification tasks, Accuracy (or Error Rate) is commonly employed as the primary metric. AUC and F1 scores are further used to address imbalanced label distributions, while Expected Calibration Error (ECE) [173], [174] calculates the weighted average error of the estimated probabilities. All criteria are the higher, the better, except the error rate and ECE. For regression tasks, common metrics include Mean Squared Error (MSE), Mean Absolute Error (MAE), and Root Mean Squared Error (RMSE), with MAE and RMSE sharing the scale of the original labels. Lower values denote superior performance. Additionally, the coefficient of determination $(\mathbb{R}^2)$ is employed, with higher values indicating a better fit.
+
+In tabular machine learning, the diversity of datasets makes it difficult for any single model to consistently excel across all scenarios. Therefore, evaluating models requires not only assessing their performance on individual datasets but also employing aggregated metrics that capture their overall effectiveness across multiple datasets.
+
+Evaluation on A Set of Tasks. Early research predominantly relied on Average Rank (Friedman Rank) [12], [39], often used in conjunction with Critical Difference Comparisons, to evaluate model performance across multiple datasets. Models are ranked per dataset based on a chosen metric (e.g., accuracy, AUC, RMSE), and the average rank is computed across datasets. To ensure statistical robustness, hypothesis tests were employed to assess the significance of ranking differences, providing a more reliable comparative analysis. For multiple comparisons, tests such as the Wilcoxon-Holm, Fredman, and Nemiyi tests are employed [175]. To address the potential degradation of average rank by poor performance on some datasets, the Probability of Achieving the Maximum Accuracy (PAMA) [12] is defined as the fraction of datasets in which a model attains the highest accuracy. An alternative to PAMA accounts for near-optimal performance, $P95$ quantifies the likelihood of a model attaining at least $95\%$ of the maximum accuracy, which is computed as the ratio of datasets where the classifier achieves at least $95\%$ of the maximum accuracy to the total number of datasets.
+
+As research progressed, more diverse evaluation metrics were introduced. The Arithmetic Mean of a chosen metric provides a direct comparison across datasets, but variations in the scales of evaluation metrics across datasets can distort results. To mitigate this issue, performance metrics are often normalized before aggregation, with normalized Accuracy applied to classification tasks and normalized RMSE (nRMSE) used for regression [36], [34]. Depending on the evaluation framework, Mean Normalized Error can be used, but its dependence on normalization can hinder independent optimization. To further address these limitations, the Shifted Geometric Mean (SGM) error was introduced, which aggregates errors multiplicatively, reducing sensitivity to extreme values and ensuring more stable cross-datasets/splits comparisons [34].
+
+Beyond absolute performance, relative comparisons are also important. The Relative Improvement metric quantifies a model's performance gain over a baseline (e.g., a simple MLP), offering insight into efficiency relative to simpler alternatives [176]. More recently, drawing inspiration from the ELO rating system[177], [178], ELO-based evaluation has
+
+been introduced [179], modeling model-to-model comparisons as pairwise competitions across datasets. The ELO Score iteratively adjusts rankings based on relative performance, providing a more dynamic, fine-grained assessment.
+
+# 2.5 Tabular Benchmarks and Datasets
+
+This section introduces existing benchmarks and datasets, along with associated considerations for constructing the benchmarks and evaluation protocols.
+
+# 2.5.1 Popular Tabular Benchmarks and Datasets
+
+We first introduce several benchmarks based on raw features constructed from various aspects. Then, we present datasets with rich semantics, following some tabular toolboxes and evaluation protocols.
+
+Standard Benchmarks. Methods for tabular data have preferences depending on the dataset, and evaluating them on limited datasets can be easily influenced by randomness or other factors. Therefore, it's important to consider various aspects to ensure a more comprehensive and reliable benchmark evaluation.
+
+A comprehensive benchmark should cover a diverse set of datasets to test the model's generalization capabilities across different tasks and feature types. The benchmark should include datasets from different task types, including binary classification, multi-class classification, and regression tasks. [12] evaluates 179 classifiers across 17 families on 121 datasets, concluding that Random Forest variants were the most likely to perform best overall. [50] explores MLPs with parameterized techniques, such as ensembling and data augmentation, over 40 classification datasets. Similarly, [33] demonstrates the effectiveness of MLPs, ResNets, and transformer-based models on 11 datasets. [36] conducts experiments on 45 datasets, investigating the differences between tree-based and DNN-based methods.
+
+The benchmark should cover datasets with varying sizes, including datasets with a large number of samples and features as well as smaller datasets. The diversity of dataset sizes helps evaluate the scalability and efficiency of different models. [39] includes 176 classification datasets and evaluate 19 methods, comprising 8 classical and 11 deep methods. In this study, the pre-trained TabPFN model [89] emerges as the top performer on average, even when limited to randomly sampled training sets of 3000 examples. However, limited trials for hyperparameter tuning and strict time constraints in [39] may have led to suboptimal evaluations for some deep tabular methods [180].
+
+To ensure robustness and generalization, datasets from multiple domains should be included. Common domains for tabular data include healthcare, biology, finance, education, and physics. Additionally, some datasets are derived from other domains, such as image or speech data, by feature extraction. [181] evaluates attention mechanisms and contrastive learning methods across 28 tabular datasets, comparing their performance with traditional deep learning and machine learning approaches. [44], with a particular focus on DNN-based models, uses a benchmark of over 300 tabular datasets spanning a wide range of task types, sizes, and domains. A more diverse collection allows us to assess whether a tabular method can generalize across applications.
+
+Semantic-Enriched Datasets. In addition, recent research has also focused on evaluating tabular data with rich semantics, such as incorporating meta information related to tasks or integrating attribute names. UniTabE [182] introduces a 7TB dataset containing 13 billion tabular examples for tabular pre-training, covering domains with investing, time series analysis, finance, economics, and with numerical, categorical, text data types. CM2 [79] proposes OpenTabs for crosstab pre-training, which contains an extensive collection of large-scale tables with column name semantics, including approximately 46M tabular samples. TP-BERTa [78] filters the OpenTabs for datasets with at least 10,000 samples and no more than 32 features, resulting in 101 binary classification datasets and 101 regression datasets with about 10 million samples. GTL [81] curates a collection of 384 public tabular datasets from Kaggle, which includes 176 classification and 208 regression tasks spanning a wide range of industrial domains. TabLib collects a set of 627M tables totaling 69TiB, along with 867B tokens of context [183]. TabLib was extracted from numerous file formats, including CSV, HTML, SQLite, PDF, Excel, and others, sourced from GitHub and Common Crawl. T4 (The Tremendous Tablib Trawl) [92] takes account of the inscrutable statistics and call sheets with personally identifiable information in TabLib and filters TabLib into a collection of 4M tables with 2.1B rows.
+
+Among these benchmarks and datasets, the semantic-rich ones are primarily used for pre-training LLMs on tabular data, while the others are mainly employed for evaluating standard methods. Besides, some toolboxes implement methods over tabular data, including those for classical methods, as well as those for deep tabular methods [184], [185], [186], [187], [188]. To establish a comprehensive tabular benchmark, several factors need to be considered, including the range of datasets and data quality.
+
+Remark 3. Recent studies have proposed alternative perspectives for tabular evaluations, such as focusing on dataset age [42], leveraging expert-level feature engineering [43], and considering dataset version [44]. Studies have also highlighted generalization in open word environments in tabular datasets [43], [109], where the distributions of training, validation, and test sets differ significantly. More discussions are in Section 9. Incorporating diverse, high-quality datasets helps build a reliable benchmark for meaningful model comparisons.
+
+# 2.5.2 Evaluation Protocols
+
+Given the strong sensitivity of tabular methods to data and the additional randomness in deep methods, robust evaluation is essential. Furthermore, due to the high computational cost of some methods, it is equally important to ensure evaluation efficiency.
+
+Model Selection. Model selection on the validation set involves both hyperparameter tuning and early stopping, which are essential for reliable evaluation. Due to the large number of hyperparameters in deep methods, automated methods like Optuna [189] are commonly used to explore hyperparameters through multiple trials [33], [69]. During tuning, models are evaluated on the validation split, while models can also be trained with multiple random seeds, providing more reliable evaluations. In each trial and the
+
+final training, early stopping [190] often employed to prevent overfitting, and the epoch with the best validation performance is selected as the final model.
+
+Performance Evaluation. To assess generalization and prevent overfitting, models are typically evaluated using separate train/val/test splits, with a typical split ratio of $64\% / 16\% / 20\%$ . However, such fixed splits may yield inconsistent results. With the rise of deep learning, researchers have proposed more robust evaluation protocols to better reflect model capabilities [191]. Two main approaches are commonly used: (1) fixing the data split and running multiple trials with different random seeds [54], [59], [105], [69], [62], [87], [33], [58], [192], [65], [71]; and (2) using cross-validation, where new train/val/test splits are generated in each fold [63], [89], [193], [68], [34]. A hybrid strategy combining both random seeds and cross-validation is also adopted [194].
+
+Recent studies show that holdout-based hyperparameter tuning can be unstable and prone to overfitting to the validation set [195], [180]. [180] found it ineffective on most TabZilla [39] datasets and instead used 5-fold cross-validation for more robust hyperparameter selection. As a result, they found the key meta-feature findings reported in [39] no longer held. This observation was also discussed in [44], which further identified meta-features that have a greater impact on model performance. For small datasets, alternative strategies have been proposed [196], [197], [198]. However, this approach significantly reduces the efficiency of hyperparameter search. [199] showed that simply reshuffling data splits can often improve generalization, making holdout selection competitive with cross-validation while remaining more computationally efficient.
+
+# 3 FROM CLASSICAL TO DEEP METHOD
+
+We present possible advantages of deep learning for tabular data, as well as the potential challenges of deep learning when compared with tree-based methods.
+
+# 3.1 Advantages of deep representation learning
+
+Deep tabular models offer several advantages beyond performance when compared with classical methods.
+
+Ability to Model Complex Feature Interactions. DNNs are particularly adept at capturing high-order, non-linear interactions between features, which may be challenging for traditional models like linear regression or decision trees [51], [54]. By learning a hierarchical representation of features, DNNs allow low-level feature interactions to be captured in the initial layers, while higher-order interactions are identified in deeper layers. This ability to automatically learn complex relationships makes DNNs highly effective in capturing intricate dependencies within tabular data.
+
+End-to-End Learning. Unlike traditional machine learning methods, which often involve separate steps for feature engineering, preprocessing, and model tuning, DNNs can process raw features and automatically extract useful representations without complex manual transformations. This end-to-end learning approach reduces human bias and simplifies the workflow, making the process more efficient. DNNs are trained through gradient optimization, enabling
+
+a unified, streamlined solution for complex tasks [33], [107]. Additionally, deep models support multi-task learning, allowing related tasks to benefit from shared representations, enhancing both performance and efficiency [200], [70], [49]. Integration with Other Modalities. Deep tabular methods excel in multi-modal pipelines, where tabular data is integrated with other modalities, such as images, audio, or text. In AI4science applications, for instance, tabular data might be combined with image data [106], [107] (e.g., in medical imaging applications) or time-series data [201], [202] (e.g., in forecasting tasks). DNNs are well-suited to model interactions between heterogeneous data types, improving the overall performance. By jointly learning from multiple data sources, DNNs enhance their ability to make more accurate and comprehensive predictions across domains.
+
+Flexibility with Dynamic Environments. DNN-based methods benefit from the flexibility of gradient-based optimization, which allows efficient and iterative training. This flexibility makes DNNs adaptable to changing objectives without significant modifications, unlike tree-based models that often require specialized methods for different tasks [9]. Moreover, DNNs excel in dynamic environments, such as real-time predictions, financial analysis, and decision-making systems, where feature relationships may shift. This adaptability makes them suitable for online learning or incremental training, where new data is continuously integrated without retraining from scratch [203], [204].
+
+Long-Term Knowledge Transfer and Learning. DNNs are capable of long-term learning and knowledge transfer, which allows them to retain valuable knowledge gained from training on diverse tasks [205]. Once trained on a broad set of tasks, DNNs can transfer this knowledge to related domains, reducing the need for complete retraining [206]. This is especially advantageous in fields like AI4science, where a model trained on one type of scientific data can be adapted to other related domains, saving both time and computational resources. This ability to transfer knowledge across tasks is a key advantage of deep learning, enabling more efficient use of data and model capabilities over time.
+
+# 3.2 Debates between Tree-Based Methods and DNNs
+
+Although deep tabular methods have shown great potential in learning semantic representations and constructing nonlinear predictors, their initial performance often struggles to surpass that of classical tree-based ensemble methods, such as Gradient Boosted Decision Trees (GBDT). Many studies still treat GBDT approaches as strong baselines [36], [39], and in some cases, the advantages of deep tabular methods diminish as the number of evaluation datasets increases.
+
+Several reasons contribute to why tree-based methods retain their advantages over DNNs in many tabular tasks:
+
+Better Handling of High-Frequency Data. Tree-based methods, particularly GBDT models, are highly efficient at handling high-frequency data or dense datasets with many small variations [38]. These models build decision trees by recursively splitting the data at the most informative feature points, capturing both local and global patterns efficiently. DNNs, on the other hand, may not capture fine-grained patterns as effectively without extensive regularization or tuning [207], [208]. To address this limitation, [38] introduced frequency reduction as an inductive bias through
+
+the addition of scaling layers, while [45] demonstrated that periodic activation functions can significantly enhance neural networks' ability to learn high-frequency functions.
+
+Natural Handling of Mixed Data Types. Tabular data often includes a combination of numerical, categorical, and ordinal features [9], [44], [209]. Tree-based models are particularly strong when working with mixed data types, as they can handle categorical features directly without requiring one-hot encoding or embeddings. This ability to work with raw categorical data simplifies the preprocessing pipeline significantly. DNNs, however, generally require encoding techniques (e.g., one-hot encoding or learned embeddings) for categorical features, adding complexity and potentially leading to suboptimal performance [63].
+
+Lower Computational Requirements for Training and Inference. For certain tasks, tree-based models tend to be more computationally efficient than DNNs [33]. GBDTs and other decision tree-based models can train relatively quickly and are less computationally intensive than deep neural networks [210], [39]. This is especially true when the dataset is not massive or when the model needs to be trained and deployed rapidly. DNNs, on the other hand, often require significant computational resources (e.g., GPUs, longer training times) to achieve comparable performance, making them less ideal in resource-constrained environments [211], [88].
+
+Robustness to Noisy and Missing Data. Tree-based models are generally more robust to noisy data and missing values. When training a decision tree, missing values can be handled through optimal splitting that accommodates absent data, and trees can effectively deal with noisy or inconsistent data points [36]. DNNs, in contrast, are more sensitive to noise and often require careful preprocessing or specific techniques (e.g., data imputation or noise filtering) to avoid performance degradation with noisy or missing data [65], [89].
+
+Interpretability and Transparency. Tree-based models offer a significant advantage in terms of interpretability [60], [61], [105]. The decision-making process of models like GBDT can be easily visualized in the form of decision paths, and feature importance can be directly extracted [130], [132], [131]. This transparency makes tree-based models appealing in domains where model explainability is crucial, such as in finance, healthcare, and regulated industries. Although interpretability techniques like LIME [212] and SHAP [213] exist for DNNs, tree-based models still tend to be more intuitive and easier to explain, especially in complex decision-making environments. Recent works [214], [60], [59], [193] have sought to bridge this gap by enhancing neural network interpretability through emulation of tree-based model behaviors.
+
+Handling Outliers and Skewed Data. Tree-based methods are often better at handling outliers and skewed distributions in the data. When a feature exhibits extreme values or skewed distributions, decision trees are inherently less sensitive to such anomalies because they create splits based on feature ranges that naturally isolate outliers. This characteristic can make them more robust than DNNs, which may require specialized loss functions or techniques (e.g., robust scaling or outlier removal) to handle such data points [43], [109].
+
+# 4 TAXONOMY OF SPECIALIZED METHODS
+
+Similar to the evolution of deep learning, which progresses from specialized learning to transfer learning and ultimately to foundation models [244], we categorize deep tabular methods into three groups, as shown in Figure 2: specialized methods, transferable methods, and general methods. This classification reflects both the evolutionary development of deep learning techniques and the increasing generalization capabilities of these models.
+
+Specialized methods, being the earliest developed and most widely used category, will be our starting point for discussion. Tabular data consists of features (columns), samples (rows), and objectives (labels), which together define the structure and the task objectives. We emphasize detailed strategies for obtaining high-quality representations at both feature- and sample-level for the target task. Specifically, given the input data, according to the general learning objective in Equation 1, we consider how to transform the tabular input $x_{i}$ (feature aspect), how to construct relationships between samples (sample aspect), how to design the objective $\ell(\cdot)$ and regularize $\Omega(\cdot)$ (objective aspect). In particular,
+
+- Feature Aspect. We focus on how to transform the raw tabular input (in various forms) into intermediate representations. We consider two types of features: numerical and categorical. By explicitly modeling the relationships between the two features (e.g., feature importance and interactions), we are able to enhance the model's understanding of the input space.
+
+- Sample Aspect. In addition to features, we explore how to retrieve and utilize neighboring samples to capture intersample dependencies, thereby improving predictions. In order to improve the model's ability to make predictions, we explore the relationships between a target sample and its "extracted neighbors."
+
+- Objective Aspect. We examine how to modify the loss function and overall objective to introduce inductive biases. By directly guiding the learning process with the target variables, we incorporate prior knowledge or task-specific preferences into the model, thereby improving its generalizability and interpretability.
+
+In specialized methods, we focus solely on learning from pure data, excluding feature semantics considered in transferable methods (in Section 6), as they leverage the capabilities of language models. Since specialized methods encompass a wide range of approaches, and feature-aspect methods are the most extensive part of them, we will first introduce sample-aspect methods and objective-aspect methods in the following subsections. In Section 5, we will provide a detailed introduction to feature-aspect methods.
+
+# 4.1 Sample-aspect Specialized Methods
+
+Sample interaction methods take a retrieval-based approach, focusing on relationships between individual samples rather than features. In a tabular dataset, each sample $x_{i}$ represents a row with $d$ features, and the goal is to leverage relationships between a target sample and its "extracted neighbors" to improve predictions.
+
+The general form for the sample interaction methods can be expressed as:
+
+$$
+\hat {y} _ {i} = f \left(\mathcal {R} \left(\boldsymbol {x} _ {i}, \mathcal {D}; \Phi\right)\right), \tag {2}
+$$
+
+Table 1: The taxonomy of representation learning for tabular data. The shade color in the last column denotes the subcategory, which is consistent with Figure 3.
+
+| Algorithm Category | Reference |
| Specialized Methods | § 5Feature-aspect Methods | Feature Encoding | [33], [45], [64] |
| Feature Selection | [59], [60], [105], [61], [193] |
| Feature Projection | [52], [33], [34], [58] |
| Feature Interaction | [54], [62], [63], [55], [65], [49], [215] |
| § 4.1Sample-aspect Methods | Sample Interaction | [70], [216], [217], [192], [67] |
| Neighbor Retrieval | [218], [68], [69], [35] |
| § 4.2Objective-aspect Methods | Training Objective | [67] |
| Training Regularization | [219], [50], [66] |
| § 6Transferable Methods | Homogeneous | [63], [48], [70], [220], [46], [221], [222], [223], [47], [224], [225], [226], [227] |
| Heterogeneous | [228], [229], [222], [72], [73], [64], [230], [231] |
| Language Model | [77], [232], [182], [79], [78], [233], [234], [82], [83], [235], [236], [80], [237] |
| Vision Model | [238], [239], [240], [74], [75], [241], [242], [76] |
| § 7General Mehtods | Raw-Feature-based | [86], [87], [88] |
| TabPFN Variants | [89], [91] |
| Semantics-based | [92], [93], [94], [243] |
+
+
+Figure 3: The roadmap of deep representation learning tabular methods. We organize representative methods chronologically to show the concentration at different stages. Different colors of these methods denote the sub-categories.
+
+where $\mathcal{D}$ is the set of all samples (training data) available for retrieval or learning. $\mathcal{R}(\cdot)$ is the sample interaction module, which retrieves or aggregates information from relevant samples in $S$ for the target sample $\boldsymbol{x}_i$ . $\Phi$ represents the learnable parameters of $\mathcal{R}$ . $f(\cdot)$ is the prediction head that maps the aggregated information to the final output $\hat{y}_i$ .
+
+Sample aspect approaches can be broadly categorized into two main strategies. The first approach introduces the modeling of sample relationships $\mathcal{R}$ during representation training, allowing the model to learn better representations by capturing inter-sample dependencies. The second ap
+
+proach is retrieval-based models, which directly predict outcomes by learning how to retrieve and utilize neighbors' relationships $\mathcal{R}$ when testing.
+
+Sample Interaction. These methods assist in representation learning by allowing the model to capture relationships between samples, which in turn helps generate a more robust representation during training. During testing, the model becomes more sensitive to each sample without interaction.
+
+SAINT [70] introduces inter-sample attention beyond inter-attribute attention, which improves row classification by relating each row to others in the table. NPT [216]
+
+extends this via non-parametric Transformers, whereas Hopular [217] employs Hopfield networks, sharing conceptual alignment with SAINT [70]. Unlike nearest-neighbor classification, the distance metric is learned end-to-end. Prompt [192] posits that the feature importance in tabular data is sample-dependent. During feature extraction, it treats the information between samples as prompts. PTaRL [67] identifies two issues in the representation of tabular data samples: entanglement and localization. It addresses these by modeling global sample relationships through prototype generation and representation projection, helping the model produce clear and consistent decisions.
+
+Neighbor Retrieval. These methods construct high-quality contexts to aid prediction by retrieving valuable neighbors and designing efficient ways to utilize them based on the relationships between samples. The training data is used to assist during testing.
+
+DNNR [68] argues that a key advantage of neighbor-based methods is the model's transparency, meaning that the model's decisions can be explained by inspecting its components. It enhances predictive performance by incorporating local gradient estimation and Taylor series approximation into the KNN framework. TabR [69] proposes that, compared to purely parametric (e.g., retrieval-free) models, retrieval-based models can achieve superior performance while also exhibiting several practically important properties, such as the ability for incremental learning and enhanced robustness. It encodes all candidate samples and then employs an attention-like mechanism to retrieve the samples that aid in making predictions, as explored in [218]. ModernNCA [35] revitalizes the classic tabular prediction method, Neighbourhood Component Analysis (NCA) [245], by designing and incorporating deep learning architectures and strategies. The resulting method efficiently leverages neighboring samples for prediction.
+
+Remark 4. The neighborhood-based approach closely resembles
+
+bles the current in-context learning [246] mechanism. In particular, the in-context learning used in general models like TabPFN [89], [91] can aslo be considered a form of the neighborhood method. This concept of neighborhood not only helps in standard tasks, but also enhances transferable and general methods. For example, LoCalPFN [90] highlights that employing local linear regression can lead to more expressive decision boundaries, while utilizing local context allows performance to scale with the size of the training dataset.
+
+# 4.2 Objective-aspect Specialized Methods
+
+The general objective learning $f$ follows the structural risk minimization as in Equation 1, where $\ell$ is the loss function to set the training objective between the prediction and the ground truth label. $\Omega(\cdot)$ is the regularization on the model, which directs the objective or restricts the complexity of $f$ .
+
+In traditional machine learning, models often rely on explicit regularization techniques on $\Omega$ to ensure good generalization. Methods such as decision trees, support vector machines, and linear models typically incorporate regularization terms directly into the loss function to control model complexity and prevent overfitting. For example, in linear regression, regularization methods like L1 (Lasso) [247], L2
+
+
+
+
+Figure 4: Illustration of feature-aspect methods, including feature encoding, feature selection, feature projection and feature interaction.
+
+
+
+
+
+(Ridge) [248], or Elastic-Nets [249] penalize large coefficients, effectively controlling the complexity of the model and helping to maintain a balance between bias and variance.
+
+Objective-aspect methods in deep learning are an extension of these traditional regularization techniques, where inductive bias is introduced by adjusting the loss function $\ell$ or adding regularizers $\Omega$ . In the training process, the goal is to leverage regularization on the model to improve predictions.
+
+Remark 5. Pre-train methods such as homogeneous transferable tabular methods in Section 6 also change the loss function $\ell$ or the regularization $\Omega$ to help pre-training. We will discuss these methods later.
+
+Objective-aspect approaches can be broadly categorized into two main strategies. The first approach involves training objectives, which enhance the model with a specialized ability. The second approach introduces a regularizer, allowing the model to learn strong generalized representations.
+
+Training Objective. For training objectives, PTaRL [67] constructs prototype-based projection space and learns the disentangled representation around global prototypes. PTaRL uses a diversification constraint for representation calibration and introduces a matrix orthogonalization constraint to ensure the independence of global prototypes.
+
+Training Regularization. For training regularization, RLNs [219] overcome the challenge of an intractable number of hyperparameters during training by introducing an efficient tuning scheme, which minimizes a new "Counterfactual Loss." In RLNs, the regularization coefficients are optimized together with learning the network weight parameters. RLNs produce extremely sparse networks, thus providing more interpretable models and revealing the importance that the network assigns to different inputs. [50] introduces "cocktails," dataset-specific combinations of 13 regularization techniques, showing that even simple neural networks can outperform tree-based architectures when optimized with these methods. TANGOS [66] introduces a regularization-based improvement. It regularizes neuron attributions to encourage neurons to specialize and become orthogonal to one another.
+
+# 5 FEATURE-ASPECT SPECIALIZED METHODS
+
+Tabular data is characterized by a diverse set of features, including both categorical and numerical variables. The complexity of tabular data arises from the variety of feature types, their interrelationships, and the high dimensionality often present. Traditional methods often rely on manual feature engineering, using techniques such as encoding categorical variables and selecting relevant features to improve model performance and reduce overfitting.
+
+As deep learning has evolved, these traditional techniques have been integrated and expanded upon. Deep tabular models are capable of automatically learning complex feature representations, reducing the need for explicit feature engineering. Feature-aspect methods, such as feature encoding, selection, projection, and interaction, are essential for transforming raw tabular inputs into more informative intermediate forms. These methods help improve a model's ability to capture intricate relationships between features, thereby enhancing its generalization capabilities.
+
+# 5.1 Feature Encoding
+
+Various encoding strategies have been explored for both categorical and numerical features in tabular data. Additionally, with the advancement of the attention mechanism, feature tokenization, similar to word embeddings in natural language processing, transforms all features into embeddings.
+
+Categorical Encoding. Categorical variables represent types of data which may be divided into groups. Examples of categorical variables are race, sex, age group, and educational level [250]. The categorical features are usually transformed in an index (integer). The two most popular techniques are an Ordinal Encoding and a One-Hot Encoding.
+
+Ordinal Encoding assigns each unique category a distinct integer value. This approach is useful when the categorical variable has an inherent order, such as "low," "medium," and "high." The main advantage of Ordinal Encoding is its simplicity and efficiency, as it transforms the categorical variable into a single numeric column. However, it assumes that there is an ordinal relationship between the categories, which may not always be the case. For instance, if the categorical variable represents "color" with categories such as "red," "blue," and "green," applying Ordinal Encoding would introduce an artificial order that does not reflect any meaningful ranking.
+
+On the other hand, One-Hot Encoding creates a new binary column for each unique category in the original categorical variable. For example, for a variable "color" with three categories (red, blue, and green), One-Hot Encoding would generate three binary columns: "is_red," "is_green," and "is_green," encoding red as $(1,0,0)$ , blue as $(0,1,0)$ and green as $(0,0,1)$ . Each column indicates the presence or absence of that particular category. One-Hot Encoding is useful for nominal categorical variables, where no order exists between the categories. While One-Hot Encoding avoids the assumption of ordinal relationships, it can lead to a high-dimensional feature space if the categorical variable has many unique values, which may result in increased computational costs and potential issues with overfitting.
+
+In some cases, more advanced encoding techniques are used to address the limitations of these basic approaches.
+
+For example, Target Encoding assigns each category a value based on the mean of the target variable for that category. This method can be useful when there is a strong relationship between the categorical features and the target. In Leave-one-out embedding, every category is replaced with the mean of the target variable of that category, which excludes the current row to avoid overfitting.
+
+Numerical Encoding. For encoding, MLP-PLR [45] introduces two numerical encoding methods: Piecewise Linear Encoding (PLE) and Periodic Activation Functions. These encoding methods can be integrated with other differentiable layers (e.g., Linear, ReLU) to enhance performance. PLE produces alternative initial representations for the original scalar values and is based on feature binning. Periodic Activation Functions take into account the fact the embedding framework where all features are computed independently of each other forbids mixing features during the embedding process and train the pre-activation coefficients instead of keeping them fixed. [38] utilizes tools from spectral analysis, showing that functions described by tabular datasets often have high irregularity, and can be smoothed by transformations such as scaling and ranking to improve performance. They propose "frequency reduction" as an inductive bias during training.
+
+Feature Tokenization. Feature tokenizer performs a similar role to the feature extractor in traditional models. It transforms the input features to embeddings [62], [33]. Since the feature representations of features are very sparse and high-dimensional, a common way is to represent them into low-dimensional spaces (e.g., word embeddings).
+
+The general form for feature tokenization can be expressed as:
+
+$$
+\boldsymbol {T} _ {i, j} = \boldsymbol {b} _ {j} + \mathcal {T} \left(x _ {i, j}; \Psi\right) \in \mathbb {R} ^ {t}, \tag {3}
+$$
+
+where $\mathcal{T}(\cdot)$ is the feature tokenizer module, which transforms the input feature vector $\pmb{x}_i\in \mathbb{R}^d$ to a token embedding $T_{i,j}\in \mathbb{R}^t$ . $t$ is the dimension of token embedding. $\pmb{b}_{j}$ is the $j$ -th feature bias. $\mathcal{T}$ can be implemented with different forms. $\Psi$ represents the learnable parameters of $\mathcal{T}$
+
+In AutoInt [62], both the categorical and numerical features are embedded into low-dimensional spaces, which reduces the dimension of the input features and meanwhile allows different types of features to interact with each other. The embeddings of categorical features are computed by multiplying the embedding matrix with the multi-hot vector, while a corresponding embedding vector represents numerical features. TabTransformer [63] embed each categorical feature into a parametric embedding of dimension $t$ using Column embedding. An embedding vector is assigned to each feature, and a set of embeddings is constructed for all categorical features. Unlike TabTransformer, SAINT [70] proposes projecting numerical features into a $t$ -dimensional space before passing their embedding through the transformer encoder. FT-Transformer [33] adapts the Transformer architecture for tabular data, where all features are transformed to embeddings and applies a stack of Transformer layers to the embeddings. Specifically, the numerical tokenizer is implemented as the element-wise multiplication $\boldsymbol{T}_i^{\mathrm{num}} = \boldsymbol{b}_i^{\mathrm{num}} + x_i^{\mathrm{num}} \cdot \boldsymbol{W}_i^{\mathrm{num}}$ , and the categorical tokenizer is implemented as the lookup table $\boldsymbol{T}_i^{\mathrm{cat}} = \boldsymbol{b}_i^{\mathrm{cat}} + \boldsymbol{e}_i^T \boldsymbol{W}_i^{\mathrm{cat}}$ , where $\boldsymbol{e}_i^T$ is a one-hot vector for the corresponding categorical feature. Other transformer-based
+
+methods, like [65], [72], [230], [215], use the same feature tokenizer as FT-Transformer.
+
+# 5.2 Feature Selection
+
+The high dimensionality of tabular data often causes overfitting, where the model focuses on irrelevant features and neglects the important ones. Feature selection reduces the number of features, retaining only the most valuable information. This helps prevent overfitting, improves generalization, and reduces computational complexity.
+
+Traditional tree-based models facilitate automatic feature selection by evaluating the impact of each feature on the target during the construction process. Decision trees utilize metrics such as information gain or the Gini index for feature selection, while ensemble methods like random forests determine feature importance by assessing each feature's contribution [251], [252], [253]. Recently, modern deep learning methods for tabular data often mimic trees' structures for feature selection.
+
+GrowNet [59] and NODE [60] primarily mimic ensemble techniques. Inspired by GBDT, GrowNet designs a framework for building DNNs with multiple weak learners, where each learner's input consists of the original features plus the penultimate layer output from the previous learner. NODE uses a differentiable Oblivious Decision Tree as the base model, applying Bagging within each layer and Stacking across layers in a multi-layered structure. To make GAM [254] scalable and effective, NODE-GAM [61] modifies NODE to be a GAM, allowing GAM to learn quick, nonlinear jumps that better match patterns in real data.
+
+TabNet [105] and GRANDE [193] focus more on how tree models handle features. TabNet not only retains the representation learning capabilities of DNNs through self-supervised learning, but also incorporates the interpretability of tree models and the benefits of sparse feature selection, with a model structure designed for both feature selection and computation. GRANDE argues that the hard splits used by tree models are a key advantage over deep models, and thus proposes a method for learning hard, axis-aligned tree ensembles using gradient descent. GRANDE combines the beneficial inductive bias of axis-aligned splits with the flexibility provided by gradient descent optimization.
+
+# 5.3 Feature Projection
+
+Feature projection methods aim to project the raw data into a middle form, enhancing the representation ability for later architectures. Feature projection methods can be broadly categorized into two main approaches: MLP variants and special designed architectures. These approaches aim to enhance the model's ability to represent complex features for underlying feature structures.
+
+MLP Variants. For model architecture, RTDL [33] investigates both ResNet-like and Transformer-based architectures tailored for tabular data, proposing simple yet effective adaptations of these widely-used deep models. In particular, the MLP architecture is constructed by stacking multiple blocks consisting of Linear layers, ReLU activations, and Dropout, which transform the raw tabular features into a fixed-dimensional hidden representation. A final linear layer is then used as the classification head. The paper highlights
+
+an important insight: with proper hyperparameter tuning, even simple architectures like MLP and ResNet can achieve competitive performance on tabular benchmarks.
+
+Another contemporaneous work [50] enhances the MLP architecture by equipping it with a comprehensive suite of modern regularization techniques. Instead of introducing architectural innovations, this study focuses on systematically exploring combinations of 13 different regularization methods to identify an effective "regularization cocktail" for plain MLPs. The results demonstrate two key findings: (i) a well-regularized vanilla MLP can significantly outperform many recent, specialized neural architectures designed for tabular data; and (ii) such MLPs can even surpass strong traditional machine learning models like XGBoost across a range of benchmarks. For a more comprehensive strategy, RealMLP [34] explores multiple aspects including preprocessing, hyperparameters, architecture, regularization, and initialization.
+
+Special Designed Architectures. For units, motivated by the observation that normalization techniques are prone to disturbances during training, SNN [52] proposes the Scaled Exponential Linear Unit (SELU) to improve deep models for tabular data. NAMs [255] uses exp-centered (ExU) hidden units to improve the learnability for fitting jumpy functions. BiSHop [58] uses a dual-component approach, sequentially processing data both column-wise and row-wise through two interconnected directional learning modules. They use layers of generalized sparse modern Hopfield layers, a sparse extension of the modern Hopfield model with learnable sparsity.
+
+# 5.4 Feature Interaction
+
+Feature interaction methods aim to model relationships among features to enhance the representation power of deep learning models on tabular data. In tabular datasets, each sample $\boldsymbol{x}_i \in \mathbb{R}^d$ is described by $d$ features, and the goal is to transform these raw features into enriched representations that improve predictive performance.
+
+The general form for feature interaction methods can be expressed as:
+
+$$
+\hat {y} _ {i} = f \left(\mathcal {H} \left(\boldsymbol {x} _ {i}; \Theta\right)\right), \tag {4}
+$$
+
+where $\pmb{x}_i\in \mathbb{R}^d$ is the input feature vector for a single instance, $\mathcal{H}(\cdot)$ is the feature interaction module, which transforms the input $\pmb{x}$ by capturing feature dependencies or generating higher-order feature interactions. $\Theta$ represents the learnable parameters of $\mathcal{H}$ . $f(\cdot)$ is the prediction head that maps the transformed representation to the final output $\hat{y}$ .
+
+Feature interaction methods can be broadly categorized into two main approaches: the design of automatic feature interaction modules and the mining of implicit feature relationships. These approaches aim to enhance the model's ability to learn complex feature interactions and underlying feature structures within tabular data.
+
+Automatic Feature Interaction Modules. These methods do not assume specific feature types within the tabular dataset. Instead, they focus on improving the feature interaction process, enabling the model to learn complex, high-order feature relationships autonomously.
+
+DCNv2 [54] improves the learning of the model's feature interaction by improving the "Cross Network" structure. It
+
+employs low-rank methods to approximate feature crosses in subspaces and then integrates these subspaces using a gating mechanism. AutoInt [62] maps the original sparse high-dimensional feature vectors into a low-dimensional space and models high-order feature interactions by stacking interaction layers with a multi-head attention mechanism. Unlike AutoInt, the TabTransformer[63] only maps categorical features into contextual embeddings and feeds them into a Transformer model, while numerical continuous features are directly concatenated with the interacted contextual embeddings. When tabular data contains only numerical features, TabTransformer behaves in an MLP-like manner. Conversely, when the data contains only categorical features, TabTransformer operates similarly to AutoInt.
+
+Implicit Feature Relationships. Methods in this category typically assume that features in tabular data can be abstracted into implicit types and that it is necessary to design a suitable feature learning process to adapt to the characteristics of different types of features.
+
+DANets [55] propose the existence of underlying feature groups in tabular data, where features within each group are correlated. They learn to group input features and perform further feature abstraction. SwitchTab [49] introduces the idea of extracting sample-specific "Salient Features" and sample-shared "Mutual Information" in tabular features. It leverages self-supervised learning to assist in learning feature representations. ExcelFormer [65] argues that while DNN assigns weights to each feature, it does not actively exclude irrelevant features. To address this, it introduces Semi-Permeable Attention for feature interaction, which allows features with lower information content to access information from more informative features while preventing highly informative features from being influenced by less relevant ones. AMFormer [215] proposes the hypothesis that arithmetic feature interactions are crucial for deep tabular models. Based on the Transformer architecture, it introduces components designed to extract both additive and multiplicative interaction information.
+
+# 6 FROM SPECIALIZED TO TRANSFERABLE MODEL
+
+Instead of training a tabular model from scratch, learning based on a Pre-Trained Model (PTM) may increase the learning efficacy and reduce the resource and data requirement. For example, in a house prices prediction task, training a regressor in a certain area may benefit from a well-trained predictor from its neighborhood.
+
+Learning by reusing the PTM usually contains two stages. The first is the pre-training of a tabular model, from one or more upstream tasks. Given the PTM and a downstream task, an adaptation strategy is needed to transform the PTM to the target task or facilitate the learning of the target model. Formally, a well-trained model $g_{\Theta}$ is often available and can be leveraged to facilitate the training of $f_{\theta}$ over $\mathcal{D}$ . Here, $g_{\Theta}$ is pre-trained on a dataset $\mathcal{D}' = \{(x_j', y_j')\}_{j=1}^{N'}$ with instances $x_j' \in \mathbb{R}^{d'}$ and labels $y_j' \in [C']$ . To reuse expert knowledge in $g_{\Theta}$ , an adaptation strategy is applied: $f_{\theta} = \text{Adapt}(f_{\theta_0} \mid \mathcal{D}, g_{\Theta})$ , where $\theta_0$ is the initialization of the model. The notation could also be extended to cases with more than one PTM. The main challenge to reuse one or more PTMs is to bridge the gap between the PTM and the
+
+target tabular model [256]. We categorize PTMs into three kinds based on the source of PTM $g_{\Theta}$ .
+
+Homogeneous Transferable Tabular Model. First, the PTM may come from the same form of task (with $d' = d$ and $C' = C$ , but with different distributions $\operatorname{Pr}(\mathcal{D}') \neq \operatorname{Pr}(\mathcal{D})$ or model families $g \neq f$ ). For example, those pre-trained from other domains [71], or those unlabeled instances [48], [70].
+
+Heterogeneous Transferable Tabular Model. In addition, we consider a PTM pre-trained from a slightly different task with $\mathcal{D}$ . In addition to the previous difference, the PTM $g_{\Theta}$ may differ from $f_{\theta}$ in feature dimension $(d' \neq d)$ or target class set $(C' \neq C)$ , so the adaptation method $\mathbf{Adapt}(\cdot)$ must handle such heterogeneity [64], [230].
+
+Cross-Modal Transferable Tabular Model. Moreover, the pre-trained model could also be constructed from another modality, such as vision and language domains. The cross-modality PTM is hard to be applied to the tabular prediction task in most cases, so auxiliary information from the tabular task like the semantic meaning of attributes (i.e., the attribute names) are usually assumed to be available in this case, where PTM like large language models may provide the latent semantic meanings as external knowledge [77], [73].
+
+The main limitation of the transferable tabular model is the assumption that the data distribution of the well-trained model should be similar to the distribution of the target model. For example in the previous house price prediction task, if the PTM is pre-trained in an area distance from the target area and targets diverse problems, it is hard to utilize the PTM in the target task [222]. Since different tabular tasks may vary in their distribution, feature, or classes, the general assumption is their exist a common "dimension" between the PTM and the target task. Only the distribution changes under the shared dimension and classes, or there exists an overlap between the feature or class spaces [230]. For example, in real-world applications such as healthcare, there are numerous medical diagnostic tables. These tables usually have some features in common such as blood type and blood pressure. For rare diseases with limited data, knowledge transfer from other diagnostic tables with overlapping features becomes beneficial [228]. When the feature/label semantics are available, two different tasks may be linked through the semantic space, and textual PTMs can be used to map the tabular instance to this space or facilitate the prediction in this space [80].
+
+Pros and Cons of transferable Models. Learning with a well-trained tabular model has several advantages based on the knowledge encoded in the PTM. First, the training efficiency of the target model is improved and the model may converge fast, as the PTM may provide better initialization weights or optimization paths. Then, the target model will reduce the requirement on the data size, i.e., learning with a few-shot dataset. Training based on a PTM also reduces the number of learnable parameters, leading to parameter-efficient tuning and reducing computational resources.
+
+# 6.1 Homogeneous Transferable Tabular Model
+
+Adapting a tabular model from another domain with different distributions is investigated in the field of unsupervised domain adaptation before the era of deep learning. One representative method is the biased regularization, which
+
+
+Figure 5: Illustration of homogeneous transferable tabular methods. The pre-trained model could be constructed from supervised learning or self-supervised learning, which includes masked language model, contrastive pre-training, and hybrid methods.
+
+minimizes the difference between the weights of the PTM and the target model, i.e.,
+
+$$
+\min _ {\boldsymbol {W}} \ell (\boldsymbol {W}) + \| \boldsymbol {W} - \boldsymbol {W} ^ {\prime} \| _ {F} ^ {2} = \min _ {\Delta \boldsymbol {W}} \ell \left(\Delta \boldsymbol {W} + \boldsymbol {W} ^ {\prime}\right) + \| \Delta \boldsymbol {W} \| _ {F} ^ {2}. \tag {5}
+$$
+
+$\ell(W)$ is the loss function on the current weights $W'$ , and the regularize constraint the distance between the target model $W$ and the PTM weights $W'$ . We can reformulate the learning objective as learning the weights residual $\Delta W$ . Biased regularization can be extended to the case where $f$ and $g$ are deep neural networks such as MLP, but it fails when the target model has a different architecture with the PTM. In this case, instead of matching two models through their weights, matching their predictions also helps. For example, twice learning [253] and knowledge distillation [257].
+
+Benefiting from the strong capacity of deep neural networks, some recent studies focus on pre-training a tabular model from unsupervised instances, and then adapting the model via fine-tuning the PTM on the target (even few-shot) labeled examples. This strategy could be applied in standard supervised learning or semi-supervised learning.
+
+Supervised Pre-training Objectives. A straightforward way to incorporate the target variable into the pre-training is by using the input corruption as an augmentation for the standard supervised learning objective. [71] identifies practices to pre-train tabular deep learning models that can be universally applied to different datasets and architectures. They show that using the object target labels during the pre-training stage benefits the downstream performance and advocates several target-aware pre-training objectives.
+
+Self-Supervised Pre-training Objectives. The self-supervised pre-training objectives can be mainly categorized into three categories, including the masked language model, contrastive pre-training, and hybrid methods.
+
+Masked Language Model (MLM). MLM is the unsupervised pre-training objective, where a random subset of features is masked for each sample, and the masked values are predicted in a multi-target classification manner [63]. VIME [48] estimates mask vectors from corrupted tabular data and reconstructs feature vectors for self-supervised learning. They use the trained encoder to generate multiple augmented samples for each data point by masking each point using several different masks and then imputing the corrupted values for each masked data point. SubTab [46] finds that reconstructing the data from the subset of its features rather
+
+than its corrupted version in an autoencoder setting can better capture its underlying latent representation. SEFS [221] reconstructs the original input based on a randomly selected subset of input features, and simultaneously estimates the gate vector that defines which features are selected or not. MET [223] uses a concatenation of representations for all features instead of averaging and uses adversarial reconstruction loss in addition to the standard loss.
+
+Contrastive Pre-training. Contrastive pre-training uses data augmentations to generate positive pairs or two different augmented views of a given example, and the loss function encourages a feature extractor to map positive pairs to similar features. The key factor in contrastive learning is to generate positive and negative versions of a given instance $x_{i}$ . [70] utilizes CutMix [258] in the input space and Mixup [259] in the embedding space to obtain positive pairs, where other instances $x_{j \neq i}$ are treated as negative ones. SCARF [47] generates a view for a given input by selecting a random subset of its features and replacing them with random draws from their respective empirical marginal distributions. STab [224] relies on two (or multiple) weight-sharing neural networks with different regularizations applied to a single input. By exploiting the stop-gradient operation technique, STab can model invariance with respect to more complicated regularizations while it will not collapse to an undesired trivial solution. DoRA [226] incorporates domain knowledge, training by intra-sample pretext task and inter-sample contrastive learning to learn contextualized representations. DACL+ [220], to overcome the reliance on a particular domain, uses Mixup noise to create similar and dissimilar examples by mixing data samples differently either at the input or hidden-state levels.
+
+Hybrid Methods. [222] explores several pre-training strategies including both supervised and unsupervised ones. It considers MLM as the unsupervised pre-training objective, and sets multi-label classification as the supervised pre-training objective. By fine-tuning the PTM with several choices, including those with frozen feature extractor or not, the paper observes that supervised pre-training leads to more transferable features in the tabular domain. LFR [227] conducts pretraining by learning to simultaneously reconstruct multiple randomly generated projection functions. It considers diverse data types to show the wide-ranging applicability of learning from randomness, including tabular, vision, and language. ReConTab [225] utilizes both self-supervised learning and semi-supervised learning. It uses regularization techniques for raw feature selection and leverages contrastive learning with labels to distill the most pertinent information for downstream tasks. [71] focuses on the setup with fully labeled tabular datasets to understand if pretraining helps tabular deep learning in a fully supervised setting and compares pretraining methods to the strong supervised baselines. They show that using the object target labels during the pertaining stage is beneficial for the downstream performance and advocate several target-aware pretraining objectives. [256] provides a systematic review and summarizes the recent progress and challenges of self-supervised learning for non-sequential tabular data.
+
+
+Figure 6: Illustration of heterogeneous transferable tabular methods. During pre-training on one or multiple datasets, most of the parameters in the PTM are trained. For downstream tasks, only a small subset of parameters is fine-tuned while the rest remain fixed.
+
+# 6.2 Heterogeneous Transferable Tabular Model
+
+The main intuition lies in the mapping $f$ and $g$ work in a similar fashion, i.e., predicting the labels with similar mechanisms. Therefore, the main idea to transfer knowledge is to match the target model with the well-trained one, over the weight space or the prediction space.
+
+Early methods mainly focus on the feature-level heterogeneity between $f$ and $g$ . One main assumption is that there exists a shared set of features between the pre-trained task $\mathcal{D}'$ and the target task $\mathcal{D}$ , then we may directly copy the weights corresponding to the shared features from the PTM. Some methods extend bias regularization to deal with heterogeneous feature spaces by padding the weights with zero. OPID [260] is a one-pass learning approach, which only needs to scan each instance once and to deal with evolving streams. In the pre-training stage, OPID compresses important information of vanished features into functions of survived features, and in the adaptation stage, it is expanded to include the augmented features. ReForm [261] learns the meta-representation for each feature and based on which calculates the relationship between features in the meta-representation space. ReForm then bridges the feature space gap through optimal transport, which could be further used to transform classifiers with different features and classes.
+
+A major advantage of neural models is that they are easily fine-tuned in new domains and learn reusable features. For example, as the deep PTM has the ability to extract generalizable features for a tabular task, reusing the knowledge from the PTM can utilize the strategies designed for visual and language domains. In detail, we can fix most of the parameters in the PTM and tune the remaining parts which only have limited parameters, for example, the linear probing or parameter-efficient fine-tuning.
+
+Reuse PTM Pre-trained from One Dataset. These methods primarily focus on the difference between the pre-trained and down-streaming datasets. TabRet [72] utilizes masked autoencoding to make the transformer work in downstream tasks. To transfer pre-trained large language models to tabular tasks, ORCA [73] trains an embedder to align the source and target distributions. TabToken [64] focuses on improving the quality of the feature tokens, which are an important component in tabular deep models. TabToken leverages a conditional contrastive loss to improve the
+
+quality of learned embeddings and demonstrates enhanced transferability of deep learning models for tabular data.
+
+Pseudo-Feature method [222] utilizes pseudo-feature models individually for each new feature. In detail, given one additional feature in a downstream dataset, it first pretrains a model on the upstream data without that feature. Then Pseudo-Feature fine-tunes the pre-trained model on downstream data to predict values in the column absent from the upstream data. Next, the fine-tuned model is used back in the upstream datasets to predict and assign pseudo-values of this feature. After supplementing the upstream dataset with the "unseen" feature in the downstream task, PseudoFeature pre-trains and transfers the feature extractor to the downstream task again. This method is computationally expensive in our broader feature space adaptation scenario. Reuse PTM Pre-trained from Multiple Datasets. XTab [230] aims to enhance the transferability of the transformer. They address the challenge of inconsistent column types and quantities among tables by utilizing independent features and federated learning to pre-train the shared component.
+
+Another thread of method learns shared components such as attribute-agnostic transformation across datasets, which provides a good model initialization for partial parameters given a downstream task. [228] infers latent representations of each attribute and each response from a few labeled instances using an inference network. The attribute and response representations are enabled make predictions based on the task-specific properties of attributes and responses even when attribute and response sizes are different across tasks. DEN [229] uses a three-block architecture: a covariate transformation block followed by a distribution embedding block and then a classification block. They provide theoretical insights to show that this architecture allows the embedding and classification blocks to be fixed after pre-training on a diverse set of tasks. Meta-Transformer [231] leverages a frozen encoder to perform multimodal perception without any paired multimodal training data. In Meta-Transformer, the raw input data from various modalities are mapped into a shared space in meta learning [262], allowing a subsequent encoder with frozen parameters to extract high-level semantic features.
+
+# 6.3 Reusing a Pre-trained Language Model
+
+In some cases, the semantic meaning of features is available, making it natural to leverage pre-trained language models for tabular data. Typically, two types of semantic information can be derived from a tabular dataset $\mathcal{D}$ . First, attribute names for each of the $d$ features, $\mathcal{A} = A_{1},\ldots ,A_{d}$ , provide useful context. Additionally, meta-information such as a textual description, denoted as meta_description, can further enhance understanding. The learning process is then formulated as:
+
+$$
+\hat {y} _ {i} = f \left(\boldsymbol {x} _ {i}, \mathcal {A} \mid \mathcal {D}, \text {m e t a} _ {\text {d e s c r i p t}}\right) \tag {6}
+$$
+
+where the semantic information bridges the gap between feature spaces and facilitates knowledge transfer from pretrained tasks to downstream applications.
+
+Although pre-trained language models have demonstrated success in various domains, their application to tabular data remains limited due to the prevalence of numerical values and the scarcity of textual descriptions.
+
+
+Figure 7: Illustration of transferable tabular methods with a language model. The language model can be applied at various stages, including feature tokenization, feature engineering, and textual serialization.
+
+Moreover, concerns about data privacy and security may further restrict access to semantic information. Consequently, language models are typically applied to tabular datasets only when textual context is sufficiently available.
+
+Language Models for Feature Tokenization. When the feature space changes, language-based methods assume that semantic relationships exist between feature descriptions and rely on large-scale language models to capture these connections. For example, the feature "occupation" in one task may share semantic similarity with the feature "organization" in another, allowing feature-label relationships to be reused across different datasets. By extracting feature embeddings (tokens), tables of varying sizes can be transformed into a standardized set of tokens in a shared space. A pre-trained transformer then encodes transferable knowledge, aiding the fine-tuning process for downstream tasks.
+
+TransTab [77] trains a tokenizer based on the words present in tabular data and incorporates both column descriptions and table cells as raw input to a gated transformer model. The model is pre-trained via self-supervised learning or supervised contrastive loss and is validated on tasks such as transfer learning and feature incremental learning. PTab [232] adopts a similar approach, learning contextual representations from multiple tokenized tabular datasets before fine-tuning for downstream tasks. UniTabE [182] encodes and fuses information from column names, data types, and cell values into a set of tokens, applying an encoder-decoder architecture with Transformer and LSTM components. It is pre-trained using Multi-Cell-Masking and contrastive learning, where a sub-vector of an instance is treated as a positive sample while other instances or their subsets are considered negatives.
+
+CM2 [79] introduces a cross-table pre-training framework that integrates attribute names and feature values. CM2 uses transformers to process feature tokens and employs a prompt-based Masked Table Modeling (pMTM) self-supervised objective, where column names act as prompts to assist in predicting masked features. TP-BERTa [78] follows a similar approach but incorporates numerical discretization strategies and magnitude tokenization for feature encoding, fine-tuning smaller pre-trained language models such as RoBERTa [263] for tabular data prediction. Its pre-training objective includes supervised loss and magnitude-aware triplet loss as a regularizer.
+
+CARTE [233] utilizes a graph representation of tabular
+
+data to handle heterogeneous feature spaces, transforming textual information from column names and entries into embeddings. A graph-attentional network is then applied to contextualize entries with column names and neighboring entries. CARTE is pre-trained on the YAGO3 knowledge base [264] by constructing graphlets for tabular data and employing contrastive loss, where the original graphlet and one truncated variant are positives, while other graphlets in the batch serve as negatives. The pre-trained CARTE model is subsequently fine-tuned for downstream tasks.
+
+Language Models for Feature Engineering. Discriminative features enhance the effectiveness of subsequent tabular learning models. Binder [234] identifies task input components that are not directly answerable by a model and leverages LLMs to generate auxiliary features, particularly for knowledge grounding tasks. Given that discriminative features are often manually designed, CAAFE [265] explores the use of LLMs to generate auxiliary features based on task and feature semantics. The quality of these features is then evaluated using a general tabular model, TabPFN [89]. FeatLLM [266] enhances feature generation by incorporating example-based prompting, enabling LLMs to create new features based on textual descriptions. TaPTaP [235] is expected to capture a generic tabular data distribution after ongoing pre-training on a large-scale corpus of real-world tabular data, generating high-quality synthetic tables to support various applications on tabular data.
+
+Language Models for Textual Serialization. A direct approach to incorporating pre-trained language models involves converting tabular data into a textual format, allowing LLMs to infer relationships between features and labels based on embedded expert knowledge. This concept has been validated in semantic parsing tasks [267], [268]. LIFT [236] and TabLLM [80] serialize tabular data by integrating feature names into text and combining them with task descriptions. This enables LLMs to treat tabular prediction tasks as text generation problems. LIFT fine-tunes models on the entire training set, while TabLLM employs few-shot learning for fine-tuning. UniPredict [237] constructs prompts using metadata, sample serialization, and task instructions, fine-tuning LLMs with confidence-weighted augmented labels predicted by an external model. The approach is validated on multiple in-distribution datasets.
+
+Despite their advantages, textual serialization methods face challenges when the number of features increases, as prompts may become too large to fit within the model's context window. The effectiveness of LLMs in tabular data tasks remains constrained by the availability of semantic information and the capabilities of external tabular models. Further exploration of LLM-based methods will be discussed in the general tabular models in Section 7.
+
+# 6.4 Reusing a Pre-trained Vision Model
+
+Given the success of deep neural networks (DNNs) in visual tasks, it is intuitive to leverage the strong recognition capabilities of pre-trained vision models for tabular data. Additionally, data augmentation strategies commonly used in image processing can be introduced after transforming tabular data into a visual format. Similar ideas have been explored in time series forecasting [269] and irregular time series classification [270].
+
+
+Figure 8: Illustration of transferable tabular methods with a vision model. Tabular data can be transformed into images through dimensionality reduction, table reorganization, and the use of image markers.
+
+The primary challenge lies in representing tabular instances in an image-compatible format. In natural images, neighboring pixels often share semantic relationships, whereas tabular data lacks inherent spatial structure. Features in a tabular instance are permutation-invariant, meaning that exchanging their order does not alter the instance's meaning. Various methods have been proposed to transform tabular data into visual representations, enabling the application of pre-trained vision models fine-tuned for tabular tasks. This subsection highlights different transformation strategies that transfer tabular datasets into images.
+
+Dimensionality Reduction Transformation. Visualization strategies for tabular data naturally convert tables into images by embedding high-dimensional features into a lower-dimensional space. DeepInsight [238] projects tabular data into a 2D space using t-SNE and constructs images through convex hull analysis, applying translation, rotation, quantization, and normalization. REFINED [239] employs Bayesian Metric Multidimensional Scaling to preserve pairwise distances within the low-dimensional representation, ensuring that structurally similar features remain proximate in the transformed image.
+
+Table Reorganization Transformation. A tabular dataset $\mathcal{D}$ can be treated as a matrix and represented as a single-channel image or kernel. To enable visual PTMs to recognize meaningful spatial relationships, different strategies have been developed for structuring tabular data into images. Tabular Convolution (TAC) [240] arranges data samples into zero-mean square matrices (kernels) of odd integer dimensions. These kernels are then convolved with a fixed "base image," and the resulting images are subsequently fed to a CNN for classification. Image Generator for Tabular Data (IGTD) [74] and TabEye [75] share a similar idea, generating an image for each data sample where pixel intensities correspond directly to feature values. These methods prioritize placing similar features in close proximity but struggle with high-dimensional tabular tasks. LM-IGTD [241] extends IGTD by incorporating stochastic feature generation to enhance robustness and generalization.
+
+Image Marker Transformation. Another approach involves encoding feature values as visual markers within an image. Super-TML [242] assigns feature values to predetermined positions within an image, effectively handling categorical and numerical datasets. Tab2Visual [76] normalizes tabular data and represents each instance as a row of multiple bars, each corresponding to a specific value. Each feature
+
+is assigned a unique color to enhance visual differentiation, while bar widths are proportional to feature magnitudes.
+
+By transforming tabular data into images, these methods enable the application of powerful pre-trained vision models to tabular prediction tasks, leveraging established deep learning techniques from the vision domain to enhance tabular model performance.
+
+# 7 FROM TRANSFERABLE TO GENERAL MODEL
+
+The general model (also referred to as the tabular foundation model) represents an advancement over the transferable model. It extends the generalization capabilities of a pretrained tabular model to a variety of heterogeneous downstream tabular tasks, regardless of their diverse feature and class spaces, without requiring additional fine-tuning. In other words, given a pre-trained model $g_{\Theta}$ , it can be directly applied to a downstream tabular task $\mathcal{D}$ to predict the label of a test instance $x^{*}$ as follows:
+
+$$
+\hat {y} ^ {*} = g _ {\Theta} \left(\boldsymbol {x} ^ {*} \mid \mathcal {D}\right). \tag {7}
+$$
+
+Thus, the general model shares similarities with the transferable tabular model, but with a greater emphasis on the "zero-shot" ability, aims to construct highly adaptive architectures capable of handling a wide array of heterogeneous datasets simultaneously. Importantly, it does not require an Adapt function, which further reduces the computational cost of hyper-parameter tuning. The goal of the general tabular model is to achieve better generalization on downstream tabular datasets $\mathcal{D}$ when compared to alternative strategies, such as training a tabular model directly on $\mathcal{D}$ or adapting a transferable model.
+
+Remark 6. Distinguishing between an advanced transferable tabular model, pre-trained on a wide range of heterogeneous tabular tasks, and the general tabular model can be challenging. Some transferable tabular models, based on auxiliary feature semantics, are able to predict labels for downstream test instances directly [80]. However, their prediction ability is constrained and typically applicable only in specific areas after fine-tuning [78], [233]. The general tabular model, on the other hand, is designed to handle a wider range of heterogeneous tabular tasks, sharing similar pre-training challenges with transferable models but without utilizing additional semantics. Fine-tuning a pre-trained general model is also an option for further performance improvements [93], [96].
+
+Pre-training has revolutionized domains such as vision and language [271], [84], but its adoption in tabular data remains limited due to the inherent heterogeneity of tabular datasets. Tabular datasets can vary significantly in both dimensionality (i.e., the number of columns) and the semantic meaning of each dimension, even within the same application. For example, different healthcare datasets may capture varying levels of detail and aspects of patient information. Even within the same feature entry (e.g., the $d$ -th column), the meaning can vary (e.g., "age" vs. "height"). This contrasts with vision and text data (within the same language), where different data sources typically share the same "vocabulary" (e.g., pixels, patches, or sub-words) and similar relationships between vocabulary "elements" (e.g., neighboring pixels
+
+
+Figure 9: Illustration of general methods. These methods handle inherent heterogeneity by improving the model's adaptability or homogenizing the diverse tabular formats. Once pre-trained, they can be directly applied to downstream tasks without fine-tuning.
+
+
+
+often share colors). The lack of shared vocabulary and relationships in tabular data makes it challenging to jointly train a model across multiple datasets, let alone apply a pre-trained model directly to new downstream tasks.
+
+There are two main strategies to address the inherent heterogeneity in tabular datasets: improving the model's adaptability or homogenizing the diverse tabular formats. We categorize general tabular models into three parts based on their strategies for achieving generalizability. The first focuses on raw-feature-based approaches, among which TabPFN variants represent a rapidly evolving branch and are thus discussed separately. The third category encompasses semantic-based methods that leverage attribute and task semantics to unify heterogeneous tasks.
+
+# 7.1 Raw-Feature-based General Models
+
+To adapt a general tabular model to heterogeneous tabular datasets during the pre-training and fine-tuning stages, two main strategies can be used from the data-centric and model-centric perspectives. From the data-centric perspective, the general model may standardize tabular datasets into a homogeneous form. For instance, TabPTM [86] transforms all datasets into a uniform format using meta-representation to enable pre-training. The pre-trained model can then be applied directly to a downstream dataset or fine-tuned without introducing additional parameters.
+
+Alternatively, from the model-centric perspective, the general model may improve adaptability by tailoring it to specific tabular tasks. HyperFast [87] adopts the concept of a Hyper Network [272] in meta-learning [273], where a mapping from the tabular dataset to the weights of a classifier is learned. This mapping can then be used to predict labels for unseen instances from the task. To address datasets with varying dimensions, HyperFast projects datasets into a fixed size using random projections. To overcome the slow weight generation speed, MotherNet accelerates HyperFast by modifying its architecture with Transformer-like modules [88].
+
+# 7.2 TabPFN Variants
+
+The TabPFN family of models [89], [91] leverages the incontext learning capabilities of transformers, directly predicting labels by adapting test instances according to the context of training examples. In the first version of TabPFN, an instance $\boldsymbol{x}_i$ is padded to a fixed dimension (e.g., 100), and the features are projected to a higher dimension (e.g., $d'$ ) for further processing. The label $y_i$ is processed similarly and
+
+added to the instance embeddings. The embeddings of all $N + 1$ instances, including training and test instances, are formulated into a set of $N + 1$ tokens with $d'$ dimensions. These tokens are processed through several layers of a Transformer, and the output token corresponding to the test instance is further predicted using a 10-way classifier. TabPFN is pretrained over synthetically generated datasets with structured causal models (SCM) [274] and Bayesian Neural Networks (BNNs) [275], [276], enabling the strong in-context learning ability, with the best checkpoint selected based on some real-world datasets. Due to the high complexity of transformers, TabPFN is limited to small-scale tasks, with suggested sizes of $N < 1000$ , $d < 100$ , and $C < 10$ .
+
+TabPFN v2 introduces a specialized feature tokenizer to better handle heterogeneity. Specifically, each cell in the table is projected to a $k$ -dimensional vector using a shared mapping, and random position encoding vectors are added to differentiate features [277]. This results in a tensor of size $(N + 1) \times (d + 1) \times k$ when there is a single test instance. The label of each instance is processed similarly, and the mapped $k$ -dimensional token is concatenated with the instance tokens. A dummy label (e.g., the average of all labels) is used for the test instance since its label is unknown. A two-way attention mechanism is used, with each feature attending to the other features in its row and then attending to the same feature across its column [278]. The output token corresponding to the label of the test instance is further mapped to a 10-class classifier or regressor. Several improvements have been made in TabPFN v2, including increased context size ( $N < 10000$ , $d < 500$ ), automatic feature engineering, and post-hoc ensemble methods. [279] analyzes TabPFN from a bias-variance perspective, shedding light on its generalization capabilities. Various applications have also been explored, including tabular data generation [280], anomaly detection [281], data augmentation [282], and time series forecasting [283].
+
+The improvements of TabPFN (especially TabPFN v1) stem from several aspects.
+
+Pre-training Improvements. TabForestPFN [284] extends TabPFN by pre-training In-Context Learning (ICL)-transformers on a new forest dataset generator that creates unrealistic datasets with complex decision boundaries. TabDPT [179] pre-trains the architecture on real-world datasets using self-supervised learning and retrieval objectives, making it suitable for both classification and regression tasks. APT [285] is pre-trained utilizing adversarial synthetic data generated by adaptive agents, which systematically modify the underlying data-generating distribution and deliberately challenge the model with diverse synthetic datasets to enhance its robustness and generalization capabilities. TabICL [286] integrates tree-based SCMs using XGBoost [130] to model complex interactions and employs curriculum learning by progressively increasing synthetic dataset sizes. Scalable Improvements. The efficiency of TabPFN is highly sensitive to context size, prompting strategies to enhance scalability and performance [39]. These include compressing training data into a compact learned representation using sketching [287] or prompt tuning techniques [288], [289],
+
+1. Some variants of TabPFN are not considered general tabular models, especially the latter parts, as they require additional fine-tuning steps. We place them in this subsection due to their strong relationship with TabPFN.
+
+employing adaptive data selection methods to identify the most pertinent training examples for each test instance [290], [90], [179], [291], and replacing traditional quadratic attention with computationally efficient linear attention mechanisms [292] and state-space models (SSMs) [293].
+
+Adaptation Improvements. Some approaches improve TabPFN's performance on downstream tasks by adapting the context [90] or fine-tuning specific parts of the model [96], [284], [290], [289]. TabICL [286] employs a column-then-row attention mechanism to construct fixed-dimensional embeddings of rows, which are subsequently processed by a transformer like TabPFN v1 to facilitate efficient in-context learning. EquiTabPFN [294] introduces self-attention across target components, ensuring that the arbitrary ordering of target dimensions does not influence model predictions, enhancing the performance of TabPFN v1 to some extent.
+
+# 7.3 Semantics-based General Models
+
+By leveraging the semantic structure of tabular data, such as column names, heterogeneous tasks can be projected into a shared language space. This allows a single language model, pre-trained on diverse tabular datasets, to handle unseen tasks in a unified manner. TabuLa-8B [92] fine-tunes a Llama 3-8B LLM for tabular data prediction (classification and binned regression) using a novel packing and attention scheme for tabular prediction. GTL [93] transforms tabular datasets into an instruction-oriented language format, facilitating the continued pre-training of LLMs on instruction-oriented tabular data, which demonstrates strong performance in few-shot scenarios. GTL-S [295] unlocks the potential of GTL from a scaling perspective, revealing that scaling datasets and prediction tasks enhance generalization. [94] extends GTL by incorporating retrieval-augmented LLMs for tabular data, combined with retrieval-guided instruction-tuning for LLMs. MediTab [243] uses a data engine that leverages LLMs to consolidate tabular samples to overcome the barrier across tables with distinct schema. MediTab aligns out-domain data with the target task using a "learn, annotate, and refinement" pipeline, enabling the pre-trained model to infer for arbitrary tabular input in the domain without fine-tuning.
+
+# 8 TABULAR ENSEMBLE METHODS
+
+Ensemble learning is a natural way to improve the generalization ability of multiple base learners by leveraging their diversity. Classical methods such as Random Forest [127] and AdaBoost [126], [296] employ bagging and boosting, respectively, by ensembling multiple decision trees. These methods have proven effective for tabular data, as they reduce bias/variance and improve robustness [297].
+
+In deep tabular learning, ensemble methods can be categorized into two primary approaches: joint-training ensembles, where multiple sub-networks are aggregated within a single training pipeline, and post-hoc ensembles, where the predictions from multiple pre-trained deep tabular models are fused. One major challenge in ensembling deep tabular methods is computational efficiency, as training multiple deep models or sub-models can be computationally expensive and time-consuming.
+
+# 8.1 Joint-Training Ensembles
+
+Joint-training ensemble methods integrate diverse model architectures within a single training process to improve predictive performance while maintaining efficiency. These architectures often combine different types of models, such as linear and non-linear models [28] or tree-based and deep neural network-based approaches [63]. Tree-mimic methods leverage this concept by mixing predictions from multiple tree nodes to enhance robustness [60], [59], [193].
+
+To improve efficiency while maintaining predictive power, various techniques have been explored. Some approaches employ parameter-efficient ensembles, such as TabM [176], which uses MLPs as base learners and incorporates BatchEnsemble [298] to generate multiple diverse base learners efficiently. This prevents a large increase in the number of learnable parameters while maintaining model diversity. Similarly, BETA leverages pre-trained TabPFN by generating multiple base learners through additional parameter tuning [96]. Specifically, BETA learns multiple feature projections, feeding the projected training sets into TabPFN and aggregating the results while applying BatchEnsemble to reduce the number of additional learnable parameters.
+
+Some hybrid approaches, such as LLM-Boost and PFN-Boost, have been developed to integrate large language models and TabPFN with gradient-boosted decision trees [299]. In these approaches, LLMs and PFN serve as the initial base learners, and additional base learners are sequentially trained in a boosting manner. This approach leverages the strong prior knowledge from LLMs and TabPFN while maintaining the scalability of gradient-boosted decision trees.
+
+# 8.2 Post-Hoc Ensembles
+
+Post-hoc ensemble (PHE) methods involve combining multiple trained models to improve robustness and accuracy. Bagging-based ensembles are one of the most direct post-hoc strategies, where usually multiple models trained with different random seeds are aggregated [33], [69]. Although this approach improves model robustness, it incurs high computational overhead. Some recent studies have demonstrated that LLM-based methods exhibit diverse prediction behaviors compared to deep tabular models that do not utilize attribute names [94]. This difference in prediction styles enhances their complementarity, making them ideal candidates for ensemble methods.
+
+Instead of explicitly training multiple models, perturbation-based approaches create diverse predictions from the same pre-trained model. One such method applies feature permutation with TabPFN, leveraging the fact that TabPFN is not fully feature permutation-invariant [89]. A perturbation-based ensemble can be formed by randomly permuting the feature order in both the training and test sets and making predictions multiple times, generating multiple diverse predictors without additional training costs. TabPFN v2 introduces additional perturbations to enhance diversity among several key factors, including variations in feature encoding, feature quantization, categorical feature shuffling, SVD-based feature compression, outlier removal, and power transformations such as the Yeo-Johnson transformation [91]. These randomly selected transformations create diverse
+
+prediction patterns, enabling effective ensemble learning without requiring multiple separately trained models.
+
+Another post-hoc ensemble strategy employed in TabPFN v2 is the use of Portfolio-Based Ensemble, where a fixed set of TabPFN configurations is used [91]. A greedy ensemble selection technique is then applied to learn optimal weights for aggregating the predictions of different configurations [300]. By combining multiple perturbed models, this method improves generalization without excessive training costs. Some methods apply ensemble techniques to TabPFN v1 to handle large datasets. For instance, TabPFN-Bagging [96], [301] divides large datasets into multiple context groups, with the final results averaged to mitigate variance. BoostPFN [301] treats TabPFN v1 as weak learners, where each weak learner uses a subset of the training data as context. This approach allows BoostPFN to outperform standard Prior Fitted Networks (PFNs) on large datasets.
+
+# 9 EXTENSIONS
+
+In this section, we briefly introduce some extensions on deep tabular methods across different complex tasks.
+
+Clustering. Traditional clustering approaches often leverage enhanced distance metrics, such as the Gower distance [302], which is specifically designed for mixed data types, and interpretable prototypes, such as K-medoids. Recent advances in tabular data clustering have sought to integrate interpretability constraints with deep representation learning. For example, IDC [97] introduces a deep learning framework for general tabular data that predicts interpretable cluster assignments at both the instance and cluster levels. To address overlapping clusters, TableDC [98] integrates the Mahalanobis distance, which accounts for variance and correlation within the data. This method provides a similarity measure suitable for tables, rows, or columns in high-dimensional latent spaces.
+
+Anomaly Detection. Anomaly detection in tabular data is crucial for identifying subtle irregularities in structured datasets, such as fraudulent transactions or equipment failures. While classical techniques like Isolation Forest [303] and Local Outlier Factor [304] remain foundational, recent developments have incorporated various methods to capture contextual relationships in high-dimensional data. For instance, [305] introduces a method that learns mappings that maximize mutual information between each sample and the part that is masked out, capturing the structural nuances of samples from a single training class. ADBench [99] provides a comprehensive tabular anomaly detection benchmark with 30 algorithms and 57 benchmark datasets. Additionally, large language models (LLMs) have also been employed for anomaly detection in tabular data [306].
+
+Tabular Generation. Tabular data generation has become an essential tool for synthetic data creation, privacy preservation, and addressing data scarcity. Traditional methods, such as Bayesian networks or GANs, focus on mimicking marginal distributions, while recent advancements emphasize preserving complex feature dependencies and semantic consistency. For instance, tabular diffusion models [307] iteratively refine synthetic data to capture subtle correlations in high-dimensional datasets, outperforming GANs in terms of data
+
+fidelity. [308] introduces high-order structural causal information as a natural prior knowledge and offers a benchmark framework for evaluating tabular synthesis models. Despite these advances, challenges remain in balancing realism with privacy, such as avoiding identity leakage in sensitive datasets, and scaling to heterogeneous data types. Hybrid neuro-symbolic models [309] bridge this gap to provide trustworthy synthetic data for downstream tasks.
+
+Interpretability. Traditional gradient-boosted decision trees (GBDTs) inherently provide interpretability through feature importance scores and decision path visualization. Frameworks such as XGBoost [130] and LightGBM [131] quantify feature importance using metrics like split frequency and information gain. SHAP values [310] enable instance-level explanations by decomposing model predictions into feature contributions. The additive nature of GBDTs allows for partial dependence plots [311] to visualize feature effects while controlling for interactions. NeC4.5 [253], a novel decision tree algorithm that integrates the comprehensibility of decision trees with the generalization ability of neural network ensembles. By training a neural network ensemble to generate a new training set, NeC4.5 enhances decision tree performance while maintaining interpretability.
+
+Recent deep models specifically designed for tabular data have introduced novel interpretability mechanisms. For example, NAMs [255] combine some of the expressivity of DNNs with the inherent intelligibility of generalized additive models. They learn a linear combination of neural networks that each attend to a single input feature, which are trained jointly and can learn arbitrarily complex relationships between their input feature and the output. TabNet [105] uses sequential attention with learnable feature masks, where each decision step explicitly selects a subset of features via sparse masking. The aggregated feature usage across steps provides global interpretability comparable to GBDT's feature importance. Subsequent variants, such as TabTransformer [63], enhance interpretability by visualizing cross-feature attention patterns. FT-Transformer [33] combines feature tokenization with explainable attention, while NODE [60], NODE-GAM [61] and DOFEN [312] generalize ensembles of oblivious decision trees, benefiting from both end-to-end gradient-based optimization and multi-layer hierarchical representation learning.
+
+Open-Environment Tabular Machine Learning. Research on distribution shifts in tabular data starts with domain-to-domain shifts [110], which are commonly categorized based on the availability of target domain data. When target data is available, transfer learning techniques such as unsupervised domain adaptation [313] and test-time adaptation [314] are widely used. These methods adapt model parameters using test-time inputs but rely on access to target distributions, which may not always be feasible. In contrast, when target data is unavailable, a more practical but challenging scenario, methods aiming to enhance robustness and generalization, using approaches such as domain generalization [315], domain robustness [316], [317], label robustness [318] or ensemble strategies [95]. TableShift [110] provides a detailed analysis of this scenario.
+
+Beyond domain-to-domain shifts, temporal shifts are more general and complex. TabReD [109] emphasizes the inherent temporality of real-world tabular data, advocating
+
+for temporal splits for training and testing. [319] further propose a refined training protocol focusing on temporal evaluation, significantly improving generalization across models. To address temporal shifts, it's critical to incorporate temporal information [319]. Drift-Resilient TabPFN [174] models temporal shifts with a secondary SCM, which specifies changes in the primary model parameters. [319] introduce a plug-and-play temporal embedding that effectively captures trend and periodicity patterns, providing an adaptive mechanism to mitigate the impact of temporal shifts. Under temporal shift conditions, most methods experience performance degradation, while TabM [95] exhibits relative robustness [109]. However, [319] demonstrate that with the refined training protocol and temporal embedding, methods such as ModernNCA [35] can regain competitiveness.
+
+Multi-modal Learning with Tabular Data. Text, such as feature names, can be effectively utilized to enhance tabular data learning, as discussed in Section 6. Here, we focus on interactions with the image modality, e.g., in healthcare, where medical images require specialized equipment and expert knowledge, often in tabular form, for accurate diagnosis [320]. To tackle challenges like large medical datasets and high annotation costs, MMCL [106] uses a contrastive self-supervised learning framework that integrates images and tabular data. CHARMS [107] transfers expert knowledge from tabular data to images, improving image predictions even without tabular data during inference, thus reducing reliance on costly expert annotations. TIP [321] proposes a self-supervised learning strategy with a tabular encoder for incomplete, heterogeneous data and a multimodal interaction module for inter-modality representation learning.
+
+Tabular Understanding. Tabular understanding involves comprehending the information contained within a table and can be broken down into several tasks. For example, Table Detection (TD) [322], [323] refers to identifying the region of the image that contains the table while Table Structure Recognition (TSR) [324], [325] involves the identification of the rows and columns to identify individual table cells, which aims to recognize the cellular structures of tables from table images by extracting the coordinates of cell boxes and row/column spanning information. Table Question Answering (TQA) [326], [327], [112] refers to providing precise answers from tables to answer a user's question. Traditional methods, whether OCR-based [328], [329], [330] or OCR-free [331], [332], [333], [334], [335], have made significant strides in TSR and TD, which are relatively simpler tasks.
+
+More complex tasks, such as TQA, have also been the focus of considerable effort. For example, Donut [332] proposes a novel task and a synthetic document image generator to pre-train the model, reducing reliance on large-scale real document images. Monkey and TextMonkey [336], [337] utilize shifted window attention and use similarity measures to filter out redundant tokens. mPLUG-DocOwl [338] adapts mPLUG-Owl for OCR-free document understanding, while TabPedia [335] constructs low- and high-resolution vision encoders with a concept synergy mechanism for visual table understanding. [339] focuses on exploring various table representations and directly prompting LLMs to improve performance. Please refer to [112], [113] for more details.
+
+# 10 DISCUSSIONS
+
+In this section, we discuss several possible future directions for tabular machine learning, particularly in light of the significant potential demonstrated by tabular general/foundation models.
+
+The Ability to Handle Dynamic and Open Environments. Tabular models, particularly foundation models, will increasingly need to operate in dynamic, real-world environments where data evolves over time [340]. One of the key challenges is dealing with imbalanced datasets [155], where certain classes may be underrepresented, and the distribution of data may shift over time [110]. As a result, models need to adapt to these changes and continue providing accurate predictions. Additionally, the emergence of new classes in the data may require the model to evolve and update its predictions in real-time [341]. This calls for methods that ensure tabular foundation models can accommodate evolving data, handling both new classes and changing distributions effectively.
+
+The Coverage and Scope of Tabular Foundation Models. Current tabular foundation models have demonstrated strong performance on various unseen classification and regression tasks. However, several important questions remain about their capabilities. For instance, in addition to in-context learning [246], are there other prediction strategies that could be employed to further enhance the versatility and performance of tabular foundation models? Beyond classification and regression, can these models be extended to handle related tasks such as clustering, imputation, outlier detection, or even table-based question answering (QA)? Expanding the task scope could increase the model's utility in a wide range of applications. Furthermore, it is worth investigating whether there is a scaling law [342] for tabular foundation models. Currently, tabular checkpoints are relatively small compared to foundation models in other modalities, such as language or vision. Understanding the implications of scaling these models—particularly the trade-offs between model size and performance—will be crucial for their future development.
+
+Will Foundation Models Always Help? While foundation models have demonstrated impressive generalization abilities, there are inherent trade-offs. Similar to ensemble learning, a single foundation model may provide an "average" predictive ability across tasks, potentially losing specialized expertise for specific tasks. To address this, a promising approach could be the development of a "tabular model zoo" [343], [344]. In this paradigm, different pre-trained models, potentially including models from other domains, could be combined for a specific tabular task. Given a new task, suitable pre-trained models could be selected, adapted if necessary, and integrated for improved performance.
+
+Model Efficiency. In many real-world applications, tabular datasets are large and high-dimensional, posing significant challenges for both training and inference [345], [44]. One area of concern is how to handle extreme cases, such as when the data is exceptionally large or sparse. Foundation models should be able to scale effectively in these scenarios without sacrificing performance. Another issue is inference speed. In large-scale problems, timely predictions are essential, especially when deployed in real-time environments [292]. Opti-
+
+mizing the inference process is therefore critical to ensure that predictions can be made quickly on large, complex datasets. Lastly, the computational resources required for training and deploying foundation models can be substantial [346]. Optimizing resource usage through methods such as model pruning, quantization, and efficient training algorithms will be important to ensure that these models remain practical and accessible for a wide range of applications.
+
+Bridging the Gap Between Tabular Data and Other Modalities. Tabular data often coexists with other data modalities, such as images and text. One of the exciting challenges in the field is how to effectively integrate tabular data with foundation models from other domains [347]. Combining the strengths of tabular models with those of vision or language models could result in more powerful and versatile models capable of handling multimodal data. Exploring how to seamlessly integrate these modalities—whether through joint embeddings, cross-modal attention mechanisms, or other techniques—could unlock significant advances in tasks that require both structured tabular data and unstructured data sources like images or text.
+
+# 11 CONCLUSION
+
+Tabular data remains a cornerstone of real-world machine learning applications, and the advancement of deep learning has opened new possibilities for effective representation learning in this domain. In this survey, we present a comprehensive overview of deep tabular representation learning, covering its background, challenges, evaluation benchmarks, and the discussion between tree-based models and DNNs. We systematically categorize existing methods into three categories—specialized, transferable, and general models—based on their generalization capabilities. In addition, we discuss ensemble techniques, extensions, and some promising future directions, such as open-environment and multimodal tabular learning. We hope this survey serves as a valuable reference for understanding the current state of the field and inspires further progress in developing more robust and generalizable tabular learning methods.
+
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+ "content": "Abstract—Tabular data, structured as rows and columns, is among the most prevalent data types in machine learning classification and regression applications. Models for learning from tabular data have continuously evolved, with Deep Neural Networks (DNNs) recently demonstrating promising results through their capability of representation learning. In this survey, we systematically introduce the field of tabular representation learning, covering the background, challenges, and benchmarks, along with the pros and cons of using DNNs. We organize existing methods into three main categories according to their generalization capabilities: specialized, transferable, and general models. Specialized models focus on tasks where training and evaluation occur within the same data distribution. We introduce a hierarchical taxonomy for specialized models based on the key aspects of tabular data—features, samples, and objectives—and delve into detailed strategies for obtaining high-quality feature- and sample-level representations. Transferable models are pre-trained on one or more datasets and subsequently fine-tuned on downstream tasks, leveraging knowledge acquired from homogeneous or heterogeneous sources, or even cross-modalities such as vision and language. General models, also known as tabular foundation models, extend this concept further, allowing direct application to downstream tasks without additional fine-tuning. We group these general models based on the strategies used to adapt across heterogeneous datasets. Additionally, we explore ensemble methods, which integrate the strengths of multiple tabular models. Finally, we discuss representative extensions of tabular learning, including open-environment tabular machine learning, multimodal learning with tabular data, and tabular understanding tasks. More information can be found in the following repository: https://github.com/LAMDA-Tabular/Tabular-Survey."
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+ "content": "Tabular data, characterized by structured rows and columns, is one of the most prevalent data formats in real-world machine learning applications, spanning diverse domains such as finance [1], healthcare [2], education [3], recommendation systems [4], and scientific research. In particular, AI for scientific research (AI4science) has increasingly relied on tabular data, as numerous prominent datasets—such as those from genomics [5], chemistry [6], and climate science [7], [8]—naturally adopt tabular forms."
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+ "content": "Tabular data inherently organizes information in a structured, table-like format. In this survey, we focus primarily on supervised tabular machine learning tasks, specifically classification and regression. Beyond their structured organization, tabular datasets frequently include heterogeneous attributes [9], encompassing numerical, categorical, or mixed data types that may be either dense or sparse. Additionally, many tabular datasets present quality challenges, such as noisy measurements, missing values, outliers, inaccuracies [10], and privacy constraints [11], all of which complicate the modeling process. The most common supervised tabular tasks are classification and regression, where the goal is to learn mappings from training data to discrete or continuous targets, respectively. As illustrated in Figure 1, each row represents an instance (with its corresponding label), while each column corresponds to a specific attribute or feature [12]."
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+ "content": "Machine learning methods for tabular data have evolved significantly over the years [13], [14], [15], [16]. Recently, the rise of deep learning has profoundly impacted domains like computer vision [17] and natural language processing [18], where Deep Neural Networks (DNNs) extract semantic representations directly from raw inputs [19], [20], [21]. These learned representations have not only improved generalization but have also facilitated knowledge transfer across related tasks [22]. The flexibility of DNNs in modeling complex feature interactions and learning rich hierarchical"
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+ "content": "- J.-P. Jiang, S.-Y Liu, H.-R Cai, Q. Zhou, and H.-J. Ye are with School of Artificial Intelligence, Nanjing University, and National Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, 210023, China. E-mail: {jiangjp.liusy,zhouql.yehj}@lamda.nju.edu.cn, caihr@smail.nju.edu.cn"
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+ "content": "produced, which improve the generalization ability based on the strengths of multiple tabular models. Finally, we briefly overview other relevant extensions of tabular learning, including clustering [97], [98], anomaly detection [99], [100], [101], data generation and imputation [102], [103], [104], interpretability [63], [105], [61], multimodal learning [106], [107], open-environment tabular machine learning [108], [109], [110], [111], and tabular understanding [112], [113]. By summarizing the state of the field and identifying open challenges, we aim to guide future research and applications in tabular data representation learning."
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+ "content": "a unified, streamlined solution for complex tasks [33], [107]. Additionally, deep models support multi-task learning, allowing related tasks to benefit from shared representations, enhancing both performance and efficiency [200], [70], [49]. Integration with Other Modalities. Deep tabular methods excel in multi-modal pipelines, where tabular data is integrated with other modalities, such as images, audio, or text. In AI4science applications, for instance, tabular data might be combined with image data [106], [107] (e.g., in medical imaging applications) or time-series data [201], [202] (e.g., in forecasting tasks). DNNs are well-suited to model interactions between heterogeneous data types, improving the overall performance. By jointly learning from multiple data sources, DNNs enhance their ability to make more accurate and comprehensive predictions across domains."
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| Specialized Methods | § 5Feature-aspect Methods | Feature Encoding | [33], [45], [64] |
| Feature Selection | [59], [60], [105], [61], [193] |
| Feature Projection | [52], [33], [34], [58] |
| Feature Interaction | [54], [62], [63], [55], [65], [49], [215] |
| § 4.1Sample-aspect Methods | Sample Interaction | [70], [216], [217], [192], [67] |
| Neighbor Retrieval | [218], [68], [69], [35] |
| § 4.2Objective-aspect Methods | Training Objective | [67] |
| Training Regularization | [219], [50], [66] |
| § 6Transferable Methods | Homogeneous | [63], [48], [70], [220], [46], [221], [222], [223], [47], [224], [225], [226], [227] |
| Heterogeneous | [228], [229], [222], [72], [73], [64], [230], [231] |
| Language Model | [77], [232], [182], [79], [78], [233], [234], [82], [83], [235], [236], [80], [237] |
| Vision Model | [238], [239], [240], [74], [75], [241], [242], [76] |
| § 7General Mehtods | Raw-Feature-based | [86], [87], [88] |
| TabPFN Variants | [89], [91] |
| Semantics-based | [92], [93], [94], [243] |
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+ "content": "By leveraging the semantic structure of tabular data, such as column names, heterogeneous tasks can be projected into a shared language space. This allows a single language model, pre-trained on diverse tabular datasets, to handle unseen tasks in a unified manner. TabuLa-8B [92] fine-tunes a Llama 3-8B LLM for tabular data prediction (classification and binned regression) using a novel packing and attention scheme for tabular prediction. GTL [93] transforms tabular datasets into an instruction-oriented language format, facilitating the continued pre-training of LLMs on instruction-oriented tabular data, which demonstrates strong performance in few-shot scenarios. GTL-S [295] unlocks the potential of GTL from a scaling perspective, revealing that scaling datasets and prediction tasks enhance generalization. [94] extends GTL by incorporating retrieval-augmented LLMs for tabular data, combined with retrieval-guided instruction-tuning for LLMs. MediTab [243] uses a data engine that leverages LLMs to consolidate tabular samples to overcome the barrier across tables with distinct schema. MediTab aligns out-domain data with the target task using a \"learn, annotate, and refinement\" pipeline, enabling the pre-trained model to infer for arbitrary tabular input in the domain without fine-tuning."
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+ "content": "In this section, we briefly introduce some extensions on deep tabular methods across different complex tasks."
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+ "content": "Clustering. Traditional clustering approaches often leverage enhanced distance metrics, such as the Gower distance [302], which is specifically designed for mixed data types, and interpretable prototypes, such as K-medoids. Recent advances in tabular data clustering have sought to integrate interpretability constraints with deep representation learning. For example, IDC [97] introduces a deep learning framework for general tabular data that predicts interpretable cluster assignments at both the instance and cluster levels. To address overlapping clusters, TableDC [98] integrates the Mahalanobis distance, which accounts for variance and correlation within the data. This method provides a similarity measure suitable for tables, rows, or columns in high-dimensional latent spaces."
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+ "content": "Anomaly Detection. Anomaly detection in tabular data is crucial for identifying subtle irregularities in structured datasets, such as fraudulent transactions or equipment failures. While classical techniques like Isolation Forest [303] and Local Outlier Factor [304] remain foundational, recent developments have incorporated various methods to capture contextual relationships in high-dimensional data. For instance, [305] introduces a method that learns mappings that maximize mutual information between each sample and the part that is masked out, capturing the structural nuances of samples from a single training class. ADBench [99] provides a comprehensive tabular anomaly detection benchmark with 30 algorithms and 57 benchmark datasets. Additionally, large language models (LLMs) have also been employed for anomaly detection in tabular data [306]."
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+ "content": "Tabular Generation. Tabular data generation has become an essential tool for synthetic data creation, privacy preservation, and addressing data scarcity. Traditional methods, such as Bayesian networks or GANs, focus on mimicking marginal distributions, while recent advancements emphasize preserving complex feature dependencies and semantic consistency. For instance, tabular diffusion models [307] iteratively refine synthetic data to capture subtle correlations in high-dimensional datasets, outperforming GANs in terms of data"
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+ "content": "for temporal splits for training and testing. [319] further propose a refined training protocol focusing on temporal evaluation, significantly improving generalization across models. To address temporal shifts, it's critical to incorporate temporal information [319]. Drift-Resilient TabPFN [174] models temporal shifts with a secondary SCM, which specifies changes in the primary model parameters. [319] introduce a plug-and-play temporal embedding that effectively captures trend and periodicity patterns, providing an adaptive mechanism to mitigate the impact of temporal shifts. Under temporal shift conditions, most methods experience performance degradation, while TabM [95] exhibits relative robustness [109]. However, [319] demonstrate that with the refined training protocol and temporal embedding, methods such as ModernNCA [35] can regain competitiveness."
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+ "content": "Tabular Understanding. Tabular understanding involves comprehending the information contained within a table and can be broken down into several tasks. For example, Table Detection (TD) [322], [323] refers to identifying the region of the image that contains the table while Table Structure Recognition (TSR) [324], [325] involves the identification of the rows and columns to identify individual table cells, which aims to recognize the cellular structures of tables from table images by extracting the coordinates of cell boxes and row/column spanning information. Table Question Answering (TQA) [326], [327], [112] refers to providing precise answers from tables to answer a user's question. Traditional methods, whether OCR-based [328], [329], [330] or OCR-free [331], [332], [333], [334], [335], have made significant strides in TSR and TD, which are relatively simpler tasks."
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+ "content": "In this section, we discuss several possible future directions for tabular machine learning, particularly in light of the significant potential demonstrated by tabular general/foundation models."
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+ "content": "The Ability to Handle Dynamic and Open Environments. Tabular models, particularly foundation models, will increasingly need to operate in dynamic, real-world environments where data evolves over time [340]. One of the key challenges is dealing with imbalanced datasets [155], where certain classes may be underrepresented, and the distribution of data may shift over time [110]. As a result, models need to adapt to these changes and continue providing accurate predictions. Additionally, the emergence of new classes in the data may require the model to evolve and update its predictions in real-time [341]. This calls for methods that ensure tabular foundation models can accommodate evolving data, handling both new classes and changing distributions effectively."
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+ "content": "The Coverage and Scope of Tabular Foundation Models. Current tabular foundation models have demonstrated strong performance on various unseen classification and regression tasks. However, several important questions remain about their capabilities. For instance, in addition to in-context learning [246], are there other prediction strategies that could be employed to further enhance the versatility and performance of tabular foundation models? Beyond classification and regression, can these models be extended to handle related tasks such as clustering, imputation, outlier detection, or even table-based question answering (QA)? Expanding the task scope could increase the model's utility in a wide range of applications. Furthermore, it is worth investigating whether there is a scaling law [342] for tabular foundation models. Currently, tabular checkpoints are relatively small compared to foundation models in other modalities, such as language or vision. Understanding the implications of scaling these models—particularly the trade-offs between model size and performance—will be crucial for their future development."
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+ "content": "Will Foundation Models Always Help? While foundation models have demonstrated impressive generalization abilities, there are inherent trade-offs. Similar to ensemble learning, a single foundation model may provide an \"average\" predictive ability across tasks, potentially losing specialized expertise for specific tasks. To address this, a promising approach could be the development of a \"tabular model zoo\" [343], [344]. In this paradigm, different pre-trained models, potentially including models from other domains, could be combined for a specific tabular task. Given a new task, suitable pre-trained models could be selected, adapted if necessary, and integrated for improved performance."
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+ "content": "Model Efficiency. In many real-world applications, tabular datasets are large and high-dimensional, posing significant challenges for both training and inference [345], [44]. One area of concern is how to handle extreme cases, such as when the data is exceptionally large or sparse. Foundation models should be able to scale effectively in these scenarios without sacrificing performance. Another issue is inference speed. In large-scale problems, timely predictions are essential, especially when deployed in real-time environments [292]. Opti-"
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+ "content": "mizing the inference process is therefore critical to ensure that predictions can be made quickly on large, complex datasets. Lastly, the computational resources required for training and deploying foundation models can be substantial [346]. Optimizing resource usage through methods such as model pruning, quantization, and efficient training algorithms will be important to ensure that these models remain practical and accessible for a wide range of applications."
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