model: per-block activation recompute (--recompute)

Wrap each transformer block's forward in the checkpoint primitive when
recompute is enabled (Phase T13 / KI-3). To make the block forward a pure
segment fn (no `&self` borrow, so it can re-run in the backward closure),
extract the block body + its helpers (linear / norm_gamma / attention /
swiglu_mlp) into free functions parameterised by (cfg, compute_dtype) and add
`Block::block_params()` (the 11 leaves in the params() per-block order). The
non-recompute path calls `block_forward` directly — identical graph to before.

- `TinyTransformer::with_recompute(bool)` builder (opt-in; default off keeps the
  unchanged tape / bit-identical numerics).
- `--recompute` flag wired into bin/train and bin/train_ddp (DDP: each rank
  checkpoints independently).

Correctness gate: tests/recompute.rs builds two identical models (recompute
on/off), runs the same batched loss+backward, and asserts the forward logits,
the loss, and EVERY parameter grad match within tight fp tol — parameterised
over fp32 and bf16 (T12 composition).

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
2026-06-17 09:42:42 +08:00
parent c396b39483
commit f202351be5
4 changed files with 357 additions and 90 deletions

View File

@@ -85,6 +85,10 @@ fn main() {
// bf16 mixed precision (Phase T12): fp32 master weights, bf16 linears +
// activations. Opt-in; default fp32 reproduces v0v4 numerics.
let bf16 = args.iter().any(|a| a == "--bf16");
// Activation recomputation (Phase T13): per-block gradient checkpointing — each
// rank checkpoints its own forward/backward; exact grads, lower peak activation
// memory (lets dim1024 batch32 fit). Opt-in; default off.
let recompute = args.iter().any(|a| a == "--recompute");
let ckpt: Option<PathBuf> = args
.iter()
.position(|a| a == "--ckpt")
@@ -167,18 +171,23 @@ fn main() {
if bf16 {
println!("bf16 mixed precision: ON (fp32 master weights)");
}
if recompute {
println!("activation recompute: ON (per-block gradient checkpointing)");
}
let results = launch(
&devices,
&train_corpus,
valid.as_ref(),
&dcfg,
move |device| {
let m = build_model(cfg, device);
let mut m = build_model(cfg, device);
if bf16 {
m.with_compute_dtype(xtrain_tensor::DType::BF16)
} else {
m
m = m.with_compute_dtype(xtrain_tensor::DType::BF16);
}
if recompute {
m = m.with_recompute(true);
}
m
},
);
let r0 = &results[0];

View File

@@ -37,6 +37,16 @@ pub struct TinyTransformer {
/// `docs/11-bf16-mixed-precision.md`). The cast op's backward upcasts the bf16
/// weight grad back to fp32, so AdamW/clip/DDP stay fp32 and unchanged.
compute_dtype: DType,
/// Activation recomputation / gradient checkpointing (Phase T13, KI-3). When
/// `true`, each transformer block's forward runs through
/// [`xtrain_autodiff::checkpoint`]: the block's internal activations are NOT
/// kept on the tape during forward (only the block input is), and the block
/// forward is re-run during backward to recover them. Trades ~one extra forward
/// per block for a large drop in peak activation memory → lets dim1024 batch32
/// fit. Default `false` = the unchanged path (every activation stored), so
/// existing numerics are bit-identical; recompute is mathematically exact, so
/// grads match the non-checkpointed path within fp tolerance.
recompute: bool,
}
impl TinyTransformer {
@@ -79,6 +89,7 @@ impl TinyTransformer {
final_norm,
lm_head,
compute_dtype: DType::F32,
recompute: false,
}
}
@@ -103,16 +114,17 @@ impl TinyTransformer {
self.compute_dtype
}
/// Project `x` (activation, in the compute dtype) by weight `w` (an fp32
/// master leaf). In bf16 mode the weight is cast to bf16 via the autograd
/// `cast` op (whose backward upcasts the grad to fp32); in fp32 mode this is
/// just `matmul(x, w)`. The activation `x` already carries `compute_dtype`.
fn linear(&self, x: &Var, w: &Var) -> Var {
match self.compute_dtype {
DType::F32 => ops::matmul(x, w),
DType::BF16 => ops::matmul(x, &ops::cast(w, DType::BF16)),
_ => unreachable!(),
}
/// Enable per-block activation recomputation / gradient checkpointing (Phase
/// T13). Builder-style and opt-in; default off keeps the unchanged tape (every
/// activation stored). On, each block's forward is wrapped in
/// [`xtrain_autodiff::checkpoint`] — exact grads, lower peak activation memory.
pub fn with_recompute(mut self, recompute: bool) -> Self {
self.recompute = recompute;
self
}
pub fn recompute(&self) -> bool {
self.recompute
}
/// All learnable parameters, in a stable order. The optimizer (a hand-written
@@ -171,32 +183,36 @@ impl TinyTransformer {
h = ops::cast(&h, DType::BF16);
}
for b in &self.blocks {
// --- Attention sub-block (pre-norm + residual) ---
let normed = ops::rms_norm(&h, &self.norm_gamma(&b.attn_norm), self.cfg.eps);
let attn = self.attention(b, &normed, batch, seq);
h = ops::add(&h, &attn);
// --- MLP sub-block (pre-norm + residual) ---
let normed = ops::rms_norm(&h, &self.norm_gamma(&b.ffn_norm), self.cfg.eps);
let mlp = self.swiglu_mlp(b, &normed);
h = ops::add(&h, &mlp);
h = if self.recompute {
// Activation recomputation (T13): run the whole block forward inside
// `checkpoint` so its internal activations aren't kept on the tape;
// the block forward is re-run in backward to recover the grads. The
// segment fn captures only `Copy` config (no borrow of `self`) and
// receives the block's params via the slice, in `block_params` order.
let (cfg, cdt) = (self.cfg, self.compute_dtype);
let seg = move |x: &Var, p: &[Var]| block_forward(cfg, cdt, batch, seq, x, p);
xtrain_autodiff::checkpoint::checkpoint(seg, &h, &b.block_params())
} else {
block_forward(
self.cfg,
self.compute_dtype,
batch,
seq,
&h,
&b.block_params(),
)
};
}
let h = ops::rms_norm(&h, &self.norm_gamma(&self.final_norm), self.cfg.eps);
let h = ops::rms_norm(
&h,
&norm_gamma(self.compute_dtype, &self.final_norm),
self.cfg.eps,
);
// lm_head matmul in compute dtype. Logits stay bf16 in bf16 mode — the
// cross_entropy op upcasts to fp32 internally (no persistent fp32 logits
// buffer, a real saving at vocab 50257), and its backward casts dx back.
self.linear(&h, &self.lm_head) // [batch*seq, vocab]
}
/// A norm/QK-norm gamma in the compute dtype. fp32 master leaf → bf16 (cast
/// op, grad upcast) in bf16 mode; identity in fp32 mode.
fn norm_gamma(&self, gamma: &Var) -> Var {
match self.compute_dtype {
DType::F32 => gamma.clone(),
DType::BF16 => ops::cast(gamma, DType::BF16),
_ => unreachable!(),
}
linear(self.compute_dtype, &h, &self.lm_head) // [batch*seq, vocab]
}
/// Cross-entropy mean loss of `forward(ids)` against `targets` (`[seq]` I32).
@@ -213,68 +229,146 @@ impl TinyTransformer {
let logits = self.forward_batched(ids, batch);
ops::cross_entropy(&logits, targets)
}
}
/// Multi-head causal self-attention over a flattened batch. `x`:[batch*seq,dim]
/// (already normed), laid out sequence-major. The Q/K/V/O projections are big
/// `[batch*seq, dim]` GEMMs; the scaled-dot-product attention itself runs as a
/// fused BATCHED op over the `batch·n_heads` (sequence,head) blocks — each
/// attends within its own `[seq,seq]` causal window (NO cross-sequence
/// attention), with RoPE positions reset per sequence (`period = seq`). Causal
/// masking is applied inside the fused op's softmax kernel (no additive
/// `[seq,seq]` mask tensor).
fn attention(&self, b: &Block, x: &Var, batch: usize, seq: usize) -> Var {
let (nh, hd) = (self.cfg.n_heads, self.cfg.head_dim);
let total = batch * seq;
let bh = batch * nh;
let scale = 1.0 / (hd as f32).sqrt();
impl Block {
/// The block's learnable leaves, in the fixed order the segment forward
/// (`block_forward`) indexes them — matches the per-block slice in
/// [`TinyTransformer::params`]. This is the param order `checkpoint` passes to
/// the recompute closure.
fn block_params(&self) -> Vec<Var> {
vec![
self.attn_norm.clone(),
self.wq.clone(),
self.wk.clone(),
self.wv.clone(),
self.q_norm.clone(),
self.k_norm.clone(),
self.wo.clone(),
self.ffn_norm.clone(),
self.w_gate.clone(),
self.w_up.clone(),
self.w_down.clone(),
]
}
}
// Project, qk-norm + RoPE, then lay out as a batched [B*nh, seq, hd] tensor.
// [B*S,dim] @ [dim,dim] = [B*S,dim]
// reshape [B*S, nh, hd]
// qk-norm per-head RMSNorm over hd (Qwen3-style; Q/K only, before RoPE)
// rope [B*S, nh, hd] with per-sequence position (period = seq)
// reshape [B, S, nh, hd] → transpose(1,2) → [B, nh, S, hd] → [B*nh, S, hd]
let to_bh = |proj: Var, norm: Option<&Var>| -> Var {
let r = ops::reshape(&proj, &[total, nh, hd]);
let r = match norm {
// Per-head RMSNorm: flatten the (B*S,nh) head rows, norm over hd,
// restore. RoPE follows on the normed Q/K (mirrors xserv qwen3.rs).
Some(gamma) => {
let flat = ops::reshape(&r, &[total * nh, hd]);
let normed = ops::rms_norm(&flat, &self.norm_gamma(gamma), self.cfg.eps);
let r = ops::reshape(&normed, &[total, nh, hd]);
ops::rope(&r, self.cfg.rope_theta, seq)
}
None => r,
};
let r = ops::reshape(&r, &[batch, seq, nh, hd]);
let t = ops::transpose_4d12(&r); // [B, nh, S, hd]
ops::reshape(&t, &[bh, seq, hd]) // [B*nh, S, hd]
/// Project `x` (activation, in the compute dtype) by weight `w` (an fp32 master
/// leaf). In bf16 mode the weight is cast to bf16 via the autograd `cast` op (whose
/// backward upcasts the grad to fp32); in fp32 mode this is just `matmul(x, w)`.
fn linear(cdt: DType, x: &Var, w: &Var) -> Var {
match cdt {
DType::F32 => ops::matmul(x, w),
DType::BF16 => ops::matmul(x, &ops::cast(w, DType::BF16)),
_ => unreachable!(),
}
}
/// A norm/QK-norm gamma in the compute dtype. fp32 master leaf → bf16 (cast op,
/// grad upcast) in bf16 mode; identity in fp32 mode.
fn norm_gamma(cdt: DType, gamma: &Var) -> Var {
match cdt {
DType::F32 => gamma.clone(),
DType::BF16 => ops::cast(gamma, DType::BF16),
_ => unreachable!(),
}
}
/// One transformer block's forward: pre-norm + multi-head causal attention +
/// residual, then pre-norm + SwiGLU MLP + residual. Pure in `(cfg, cdt, batch,
/// seq, input, params)` (no `&self`) so it can be the segment fn of
/// [`xtrain_autodiff::checkpoint`] for activation recomputation (T13). `params` is
/// the block's leaves in [`Block::block_params`] order.
fn block_forward(cfg: Config, cdt: DType, batch: usize, seq: usize, h: &Var, p: &[Var]) -> Var {
let (attn_norm, wq, wk, wv) = (&p[0], &p[1], &p[2], &p[3]);
let (q_norm, k_norm, wo) = (&p[4], &p[5], &p[6]);
let (ffn_norm, w_gate, w_up, w_down) = (&p[7], &p[8], &p[9], &p[10]);
// --- Attention sub-block (pre-norm + residual) ---
let normed = ops::rms_norm(h, &norm_gamma(cdt, attn_norm), cfg.eps);
let attn = attention(
cfg, cdt, batch, seq, &normed, wq, wk, wv, q_norm, k_norm, wo,
);
let h = ops::add(h, &attn);
// --- MLP sub-block (pre-norm + residual) ---
let normed = ops::rms_norm(&h, &norm_gamma(cdt, ffn_norm), cfg.eps);
let mlp = swiglu_mlp(cdt, &normed, w_gate, w_up, w_down);
ops::add(&h, &mlp)
}
/// Multi-head causal self-attention over a flattened batch. `x`:[batch*seq,dim]
/// (already normed), laid out sequence-major. The Q/K/V/O projections are big
/// `[batch*seq, dim]` GEMMs; the scaled-dot-product attention itself runs as a
/// fused BATCHED op over the `batch·n_heads` (sequence,head) blocks — each attends
/// within its own `[seq,seq]` causal window (NO cross-sequence attention), with
/// RoPE positions reset per sequence (`period = seq`). Causal masking is applied
/// inside the fused op's softmax kernel (no additive `[seq,seq]` mask tensor).
#[allow(clippy::too_many_arguments)]
fn attention(
cfg: Config,
cdt: DType,
batch: usize,
seq: usize,
x: &Var,
wq: &Var,
wk: &Var,
wv: &Var,
q_norm: &Var,
k_norm: &Var,
wo: &Var,
) -> Var {
let (nh, hd) = (cfg.n_heads, cfg.head_dim);
let total = batch * seq;
let bh = batch * nh;
let scale = 1.0 / (hd as f32).sqrt();
// Project, qk-norm + RoPE, then lay out as a batched [B*nh, seq, hd] tensor.
// [B*S,dim] @ [dim,dim] = [B*S,dim]
// reshape [B*S, nh, hd]
// qk-norm per-head RMSNorm over hd (Qwen3-style; Q/K only, before RoPE)
// rope [B*S, nh, hd] with per-sequence position (period = seq)
// reshape [B, S, nh, hd] → transpose(1,2) → [B, nh, S, hd] → [B*nh, S, hd]
let to_bh = |proj: Var, norm: Option<&Var>| -> Var {
let r = ops::reshape(&proj, &[total, nh, hd]);
let r = match norm {
// Per-head RMSNorm: flatten the (B*S,nh) head rows, norm over hd,
// restore. RoPE follows on the normed Q/K (mirrors xserv qwen3.rs).
Some(gamma) => {
let flat = ops::reshape(&r, &[total * nh, hd]);
let normed = ops::rms_norm(&flat, &norm_gamma(cdt, gamma), cfg.eps);
let r = ops::reshape(&normed, &[total, nh, hd]);
ops::rope(&r, cfg.rope_theta, seq)
}
None => r,
};
let r = ops::reshape(&r, &[batch, seq, nh, hd]);
let t = ops::transpose_4d12(&r); // [B, nh, S, hd]
ops::reshape(&t, &[bh, seq, hd]) // [B*nh, S, hd]
};
let q = to_bh(self.linear(x, &b.wq), Some(&b.q_norm));
let k = to_bh(self.linear(x, &b.wk), Some(&b.k_norm));
let v = to_bh(self.linear(x, &b.wv), None);
let q = to_bh(linear(cdt, x, wq), Some(q_norm));
let k = to_bh(linear(cdt, x, wk), Some(k_norm));
let v = to_bh(linear(cdt, x, wv), None);
// Fused batched causal SDPA over all B*nh (sequence,head) blocks at once
// (2 batched GEMMs + 1 causal-softmax kernel; no per-head/per-seq loop).
let out = ops::attention(&q, &k, &v, scale); // [B*nh, S, hd]
// Fused batched causal SDPA over all B*nh (sequence,head) blocks at once
// (2 batched GEMMs + 1 causal-softmax kernel; no per-head/per-seq loop).
let out = ops::attention(&q, &k, &v, scale); // [B*nh, S, hd]
// Back to [B*S, dim]: [B*nh,S,hd] → [B,nh,S,hd] → transpose(1,2) →
// [B,S,nh,hd] → [B*S, dim].
let out = ops::reshape(&out, &[batch, nh, seq, hd]);
let out = ops::transpose_4d12(&out); // [B, S, nh, hd]
let concat = ops::reshape(&out, &[total, nh * hd]); // [B*S, dim]
self.linear(&concat, &b.wo) // out projection
}
// Back to [B*S, dim]: [B*nh,S,hd] → [B,nh,S,hd] → transpose(1,2) →
// [B,S,nh,hd] → [B*S, dim].
let out = ops::reshape(&out, &[batch, nh, seq, hd]);
let out = ops::transpose_4d12(&out); // [B, S, nh, hd]
let concat = ops::reshape(&out, &[total, nh * hd]); // [B*S, dim]
linear(cdt, &concat, wo) // out projection
}
/// SwiGLU MLP: `down( silu(gate(x)) ∘ up(x) )`. `x`:[batch*seq,dim].
fn swiglu_mlp(&self, b: &Block, x: &Var) -> Var {
let gate = self.linear(x, &b.w_gate); // [seq, ffn_hidden]
let up = self.linear(x, &b.w_up); // [seq, ffn_hidden]
let act = ops::swiglu(&gate, &up); // silu(gate) ∘ up
self.linear(&act, &b.w_down) // [seq, dim]
}
/// SwiGLU MLP: `down( silu(gate(x)) ∘ up(x) )`. `x`:[batch*seq,dim].
fn swiglu_mlp(cdt: DType, x: &Var, w_gate: &Var, w_up: &Var, w_down: &Var) -> Var {
let gate = linear(cdt, x, w_gate); // [seq, ffn_hidden]
let up = linear(cdt, x, w_up); // [seq, ffn_hidden]
let act = ops::swiglu(&gate, &up); // silu(gate) ∘ up
linear(cdt, &act, w_down) // [seq, dim]
}
/// Materialise a parameter's value back to a host `Vec<f32>` (for the GD step

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@@ -0,0 +1,156 @@
// T13 activation-recomputation correctness gate (the HARD gate).
//
// Gradient checkpointing is mathematically EXACT: the backward re-runs the same
// `segment_fn` from the same saved input and the same (unchanged) parameter
// values, so the recomputed activations equal the originals and the recovered
// grads equal the non-checkpointed grads — checkpointing trades compute for
// memory, never correctness. This test makes that a closed loop on-GPU:
//
// build two identical models (same init), one with `--recompute` on, one off,
// run the SAME batched loss + backward on both, and assert
// 1. the forward logits match (recompute doesn't touch forward output)
// 2. the loss matches
// 3. EVERY parameter's grad matches within a tight fp tolerance.
//
// Composition is covered by parameterising over fp32 AND bf16 (T12): the
// recompute path is the unchanged block forward, so it runs the same dtype path.
#![cfg(not(no_cuda))]
use xtrain_cuda::device;
use xtrain_model::{Config, TinyTransformer, batched_ids_tensor};
use xtrain_tensor::{DType, Device};
fn fill(n: usize, seed: u64, scale: f32) -> Vec<f32> {
let mut state = seed
.wrapping_mul(2862933555777941757)
.wrapping_add(3037000493);
(0..n)
.map(|_| {
state = state
.wrapping_mul(6364136223846793005)
.wrapping_add(1442695040888963407);
(((state >> 33) as f32 / (1u64 << 31) as f32) - 0.5) * 2.0 * scale
})
.collect()
}
fn build(cfg: Config, device: Device, dtype: DType, recompute: bool) -> TinyTransformer {
let mut seed = 1u64;
let m = TinyTransformer::new(cfg, device, |shape| {
seed = seed.wrapping_add(1);
let n: usize = shape.iter().product();
if shape.len() == 1 {
fill(n, seed, 0.02).iter().map(|v| v + 1.0).collect()
} else {
fill(n, seed, 0.08)
}
});
m.with_compute_dtype(dtype).with_recompute(recompute)
}
fn host(t: &xtrain_tensor::Tensor) -> Vec<f32> {
t.to_device(Device::Cpu).as_slice::<f32>().to_vec()
}
fn run(dtype: DType, logit_tol: f32, grad_tol: f32) {
assert!(device::device_count().unwrap() > 0, "no CUDA device");
device::set_device(0).unwrap();
let device = Device::Cuda(0);
// A few layers so checkpointing actually wraps multiple blocks.
let mut cfg = Config::tiny();
cfg.vocab = 16;
cfg.n_layers = 4;
let batch = 3usize;
let seq = 6usize;
let seqs: Vec<Vec<i32>> = (0..batch)
.map(|b| {
(0..seq)
.map(|i| ((b * 7 + i * 3 + 1) % cfg.vocab) as i32)
.collect()
})
.collect();
let tgts: Vec<Vec<i32>> = (0..batch)
.map(|b| {
(0..seq)
.map(|i| ((b * 5 + i * 2 + 2) % cfg.vocab) as i32)
.collect()
})
.collect();
let ids = batched_ids_tensor(&seqs, device);
let tgt = batched_ids_tensor(&tgts, device);
// --- recompute OFF (reference) ---
let off = build(cfg, device, dtype, false);
let off_logits = host(&off.forward_batched(&ids, batch).value());
let off_loss = off.loss_batched(&ids, &tgt, batch);
let off_loss_val = host(&off_loss.value())[0];
off_loss.backward();
let off_grads: Vec<Vec<f32>> = off
.params()
.iter()
.map(|p| host(&p.grad().expect("off grad")))
.collect();
// --- recompute ON ---
let on = build(cfg, device, dtype, true);
let on_logits = host(&on.forward_batched(&ids, batch).value());
let on_loss = on.loss_batched(&ids, &tgt, batch);
let on_loss_val = host(&on_loss.value())[0];
on_loss.backward();
let on_grads: Vec<Vec<f32>> = on
.params()
.iter()
.map(|p| host(&p.grad().expect("on grad")))
.collect();
// 1. Forward logits — recompute must not change the forward output.
let logit_rel = off_logits
.iter()
.zip(&on_logits)
.map(|(a, b)| (a - b).abs() / a.abs().max(1e-4))
.fold(0.0f32, f32::max);
// 2. Loss.
let loss_rel = (off_loss_val - on_loss_val).abs() / off_loss_val.abs().max(1e-4);
println!(
"[{dtype:?}] recompute on/off: loss {off_loss_val:.6}/{on_loss_val:.6} (rel {loss_rel:.2e}), \
logits max rel {logit_rel:.2e}"
);
assert!(
logit_rel < logit_tol,
"[{dtype:?}] logits diverged: {logit_rel:.2e}"
);
assert!(
loss_rel < logit_tol,
"[{dtype:?}] loss diverged: {loss_rel:.2e}"
);
// 3. Every parameter grad — the load-bearing gate.
let mut max_grad_rel = 0.0f32;
for (off_g, on_g) in off_grads.iter().zip(&on_grads) {
for (a, b) in off_g.iter().zip(on_g) {
let rel = (a - b).abs() / a.abs().max(1e-3);
max_grad_rel = max_grad_rel.max(rel);
}
}
println!("[{dtype:?}] recompute on/off: grad max rel err = {max_grad_rel:.3e}");
assert!(
max_grad_rel < grad_tol,
"[{dtype:?}] recompute grads diverged from non-recompute: {max_grad_rel:.3e}"
);
}
#[test]
fn recompute_matches_non_recompute_fp32() {
// fp32: recompute runs the identical deterministic kernels → grads match to
// (near) bit-exact; allow a hair for any nondeterministic GPU reduction.
run(DType::F32, 1e-5, 1e-4);
}
#[test]
fn recompute_matches_non_recompute_bf16() {
// bf16 (T12 composition): same bf16 path on recompute. The recompute is still
// exact w.r.t. the bf16 forward, so on/off match tightly (looser tol only for
// bf16 rounding, not for any recompute discrepancy).
run(DType::BF16, 5e-3, 5e-3);
}

View File

@@ -112,6 +112,10 @@ fn main() {
// bf16 mixed precision (Phase T12): fp32 master weights, bf16 linears +
// activations. Opt-in; default fp32 reproduces v0v4 numerics.
let bf16 = args.iter().any(|a| a == "--bf16");
// Activation recomputation (Phase T13): per-block gradient checkpointing —
// exact grads, lower peak activation memory (lets dim1024 batch32 fit). Opt-in;
// default off stores every activation (unchanged numerics).
let recompute = args.iter().any(|a| a == "--recompute");
let ckpt: PathBuf = PathBuf::from(
args.iter()
.position(|a| a == "--ckpt")
@@ -175,6 +179,10 @@ fn main() {
model = model.with_compute_dtype(DType::BF16);
println!("bf16 mixed precision: ON (fp32 master weights)");
}
if recompute {
model = model.with_recompute(true);
println!("activation recompute: ON (per-block gradient checkpointing)");
}
// Eval-only mode: load a checkpoint and score it on the held-out val set, then
// exit. Used to put an EXISTING model (e.g. v0) and a new one on the same